#  Published in Aerospace Journal, 2021
Numerical Investigation of a Rectangular Jet Exhausting over a Flat Plate with
Periodic Surface Deformations at the Trailing Edge

Colby Horner email<EMAIL_ADDRESS>Department of Aerospace
Engineering, Mississippi State University, MS 39762, USA;
<EMAIL_ADDRESS>Adrian Sescu email<EMAIL_ADDRESS>Department
of Aerospace Engineering, Mississippi State University, MS 39762, USA;
<EMAIL_ADDRESS>Mohammed Afsar  Department of Mechanical & Aerospace
Engineering, Strathclyde University, Glasgow, UK  Eric Collins  Center for
Advanced Vehicular Systems, Mississippi State University, MS 39762, USA

###### Abstract

Multiple competing factors are forcing aircraft designers to reconsider the
underwing engine pod configuration typically seen on most modern commercial
aircraft. One notable concern is increasing environmental regulations on noise
emitted by aircraft. In an attempt to satisfy these constraints while
maintaining or improving vehicle performance, engineers have been
experimenting with some innovative aircraft designs which place the engines
above the wings or embedded in the fuselage. In one configuration, a blended
wing concept vehicle utilizes rectangular jet exhaust ports exiting from above
the wing ahead of the trailing edge. While intuitively one would think that
this design would reduce the noise levels transmitted to the ground due to the
shielding provided by the wing, experimental studies have shown that this
design can actually increase noise levels due to interactions of the jet
exhaust with the aft wing surface and flat trailing edge. In this work, we
take another look at this rectangular exhaust port configuration with some
notional modifications to the geometry of the trailing edge to determine if
the emitted noise levels due to jet interactions can be reduced with respect
to a baseline configuration. We consider various horizontal and vertical
offsets of the jet exit with respect to a flat plate standing in for the aft
wing surface. We then introduce a series of sinusoidal deformations to the
trailing edge of the plate of varying amplitude and wave number. Our results
show that the emitted sound levels due to the jet–surface interactions can be
significantly altered by the proposed geometry modifications. While sound
levels remained fairly consistent over many configurations, there were some
that showed both increased and decreased sound levels in specific directions.
We present results here for the simulated configurations which showed the
greatest decrease in overall sound levels with respect to the baseline. These
results provide strong indications that such geometry modifications can
potentially be tailored to optimize for further reductions in sound levels.

##  1 Introduction

Methods to reduce the noise generated by aircraft propulsion systems have long
been a topic of research within the aerospace community. There have been many
challenges faced by designers attempting to solve these types of problems due
to the complexity of the flow phenomena as well as the costs involved with
testing potential solutions. In some cases, noise reductions have only come at
the expense of aircraft performance in other metrics (e.g., weight, cost).

Jet–surface interaction noise is usually associated with an increase in low
frequency noise as a result of the interaction between a turbulent jet and a
flat surface that is parallel to the jet axis. Jet noise is generally
considered to be the result of two sources: scrubbing noise and trailing edge
scattering noise (Brown and Wernet [1]). Scrubbing is generated by pressure
fluctuations in the turbulent boundary layer impinging on a flat surface.
Trailing edge scattering, which is the dominant noise source at low frequency
(Brown [2], Podboy [3]), is produced when the hydrodynamic component of the
upstream turbulence is scattered into acoustic waves after interacting with
the trailing edge. The total installation noise can propagate upstream or
downstream of the trailing edge.

High-speed jet exhausts interacting with nearby surfaces can generate flow
distortions that significantly increase the overall noise emitted. In one such
example, an aircraft engine placed on top of the wing in an attempt to shield
the jet noise that would propagate to the ground while also improving aircraft
performance actually resulted in an overall increase in sound levels. Another
example of note is the tremendous noise generated by the jet exhausts of
military aircraft interacting with the various surfaces of an aircraft carrier
deck during take-off and landing operations.

Considerable research has been conducted on axisymmetric jets (Bridges [4],
Brown and Bridges [5]), but, since the 1980s, a number of studies have shifted
their focus to non-circular jet engine exit designs as potential passive flow
control devices (Gutmark and Grinstein [6]). Geometric corrugations applied at
the nozzle exit plane (e.g., tabbed studied by Gao et al. [7], cruciform by El
Hassan et al. [8], and chevrons by Violato et al. [9]) have been observed to
break down the large-scale structures present in the jet exhaust flow. This
subsequently reduces the mixing between the jet and ambient-fluid. It was
noted that these effects were most pronounced for larger wetted perimeters of
the jet exit (Shakouchi and Iryama [10]) and an increased number of
corners/sides (Gutmark and Grinstein [6]). The effect of external expansion
ramps on supersonic flow emitted from high-aspect-ratio rectangular nozzles
was analyzed by Malla et al. [11] to determine if the ramps could be used to
reduce noise. Research conducted by Mancinelli et al. [12] and Proenca et al.
[13] show further results regarding the interaction of rectangular jets with
flat plates. Other aircraft propulsion concepts have utilized acoustically
treated engine parts (inlet ducts, exhaust ducts, and inner walls), but this
type of treatment tends to have a negative overall impact due to additional
cost and weight.

While static chevron configurations at the jet exit have been shown to provide
a significant reduction in noise without noticeable loss in thrust, some
researchers have been looking into potential acoustic benefits that could be
gained by exciting the chevrons with piezoelectric actuators. Mechanical
perturbations of the flow by the actuated chevrons are believed to increase
the production of small-scale disturbances while diminishing the large-scale
turbulent structures thought to be responsible for the dominant portion of jet
mixing noise. Mohan et al. [14] introduced piezoelectric actuators to chevrons
in an attempt to modify the growth rate of the mixing layer. Butler and
Calkins [15] examined four different types of nozzles that exist: round,
faceted, faceted with static chevrons, and faceted with active chevrons. Their
results demonstrated a 2 to 4 dB reduction in noise with the static chevrons,
and an additional 2 dB reduction with the actuated chevron configurations.

Zaman [16] studied the integration of microjets near the engine exhaust port
and found that a clear noise reduction was observed as the microjet pressure
increased. The results showed that turbulent mixing noise reduction, as
monitored by the overall sound pressure level at a shallow angle, correlated
with the ratio of the microjet to the primary jet driving pressures
(normalized by the ratio of the corresponding diameters). Semlitsch et al.
[17] used implicit large eddy simulation to analyze the relationship between
screech tone frequency and fluidic injection pressure of the microjets.

Rego et al. [18] utilized a Lattice–Boltzmann Method (LBM) to carry out
numerical simulations of a flat plate placed in an irrotational hydrodynamic
field of a jet with a round nozzle. Three cases were investigated with input
flow characteristics determined from NASA wind tunnel experiments (see Brown
[19]). Heated and cooled jets were analyzed at different Mach numbers, and the
far-field noise was computed using the Ffowcs-Williams and Hawkings equation.
Far-field spectral results showed a large noise increase for low to mid
frequencies due to upstream hydrodynamic waves in the jet that propagate along
the plate, ’scrubbing’ over the surface, until it reaches the trailing edge of
the plate. (This is referred to as the ’gust solution’ in Afsar et al. [20]).
As these waves interact with the trailing edge and the downstream turbulent
mixing layer, they are scattered in the far-field as noise. A noticeable
change in the far-field noise was observed when the trailing edge location was
changed relative to the fixed nozzle. As the length of the plate was
increased, the trailing edge was positioned in a region dominated by large-
scale structures and a greater degree of low-frequency amplification was
observed. When the plate was moved closer to the jet in the radial direction,
the sound levels at lower frequencies increased.

Behrouzi and McGuirk [21] conducted experiments at the Loughborough University
high-pressure nozzle test facility on a rectangular nozzle exhausting over a
rectangular plate acting as an aft-deck. (A configuration similar to the one
used in the current study.) The focus of the experiment was to study the
effect of the rectangular plate on flow field development and to determine if
any acoustic benefits could be obtained from the shielding effect that the
aft-deck might produce. Schlieren imaging, pneumatic Pitot probes, and
nonintrusive Laser Doppler Anemometry measurements were used to capture
detailed information on the flow structures. The data revealed that the aft-
deck created an asymmetry in the entrainment characteristics of the shear
layers, inducing a net transverse pressure force on the plume by changing the
inviscid shock cell structure. The presence of the aft-deck altered the net
pressure force exerted on the flow which led to a transverse deflection of the
jet. The aft-deck configuration produced a dramatic effect on the plume
development, extending the potential core length slightly and reducing
turbulence levels in the plume near field.

Bridges [22] further investigated this configuration by studying whether
beveled edges on the rectangular nozzle would make a difference in the sound
generated when compared to an aft-deck being added to the propulsion
configuration (see also Zaman [23] and Zaman et al. [24]). Nozzles were
fabricated with different aspect ratios of 2:1, 4:1, and 8:1. For each aspect
ratio, three nozzle designs were tried: one basic nozzle with no extension and
two beveled variants with bevel lengths of 1.3 and 2.7 inches. A rectangular
nozzle with an aft-deck surface was also considered in which the rectangular
nozzles had plates fitted to them such that when the plate surface was even
with the inner lip of the nozzle, the surface and nozzle were effectively one
piece. Five aft-deck lengths were tested: 1.3, 2.7, 4, 8, and 12 inches long.
When the beveled nozzle was analyzed, it was found that the noise levels
decreased in both azimuthal planes as the aspect ratio was increased, with
much more dramatic reductions in the nozzle’s major axis plane. These
reductions in noise were lost when one lip of the nozzle was extended to make
an aft deck. Low and high frequencies were examined with the aft deck
extensions. For low frequencies, the sound levels were roughly the same on
both sides of the plate. However, there were noticeable differences in the
sound levels for higher frequencies on each side of the plate. This is due to
the effect of the noise being shielded and reflected. The shielding effect
seemed to increase as the length of the aft deck increased, but this was not
the case for reflection. In addition, it is important to note that, for the
largest plate lengths, the low-frequency amplification above the baseline jet
noise was not dependent on the aspect ratio.

Seiner and Manning [25] considered the interaction between a supersonic jet
and a flat plate from a rectangular nozzle. They showed that the distance
between the nozzle exit and flat surface was an important parameter which can
impact the screech noise. Ibrahim et al. [26] studied the effect of turbulence
characteristics of jet flows on the radiated jet noise. Berland et al. [27]
used compressible large eddy simulations to investigate the generation of
screech tones from an under-expanded jet with a three-dimensional planar
geometry.

In the present work, we aim to extend these results by examining the effect of
sinusoidal surface deformations added to the trailing edge of the aft-deck
plate. To this end, a suite of large eddy simulations have been performed,
each targeting different geometrical configurations and flow conditions. Sound
levels in the farfield were then computed to determine whether these
deformations could potentially reduce the total noise emitted by the installed
configuration. In these simulations, we consider a high aspect ratio
rectangular jet. To simplify the nozzle geometry, we used a spanwise slice of
width four times the height of the nozzle with periodic boundary conditions
applied in the spanwise direction. Flow disturbances were imposed at the
inflow boundaries to introduce three-dimensional flow structures into the jet
flow. A high-order accurate solver was used to discretize the unsteady,
compressible, conservative form of the filtered Navier–Stokes equations. The
mean flow was determined using Reynolds Averaged Navier–Stokes (RANS)
equations via a SST $k-\omega$ turbulence model (Menter [28]). The farfield
noise radiation was evaluated using Ffowcs-Williams and a Hawkings acoustic
analogy method.

We provide further details of the numerical formulation in Section 2, and
present our results in Section 3.

##  2 Problem Formulation and Numerical Algorithms

###  2.1 Scalings

The governing equations considered here are expressed in terms of a
generalized curvilinear coordinate system, subject to the following
transformations:

$\displaystyle\xi=\xi\left(x,y,z\right),\hskip
14.45377pt\eta=\eta\left(x,y,z\right),\hskip
14.45377pt\zeta=\zeta\left(x,y,z\right),$

where $\xi$, $\eta$, and $\zeta$ are the body-aligned curvilinear coordinates
corresponding to the streamwise, wall-normal, and spanwise directions,
respectively, and $x$, $y$, and $z$ are the (non-dimensional) global Cartesian
coordinates of physical space. All dimensional spatial coordinates
($x^{*},y^{*},z^{*}$) are normalized by the reference length, $D_{j}$, the
height of the nozzle (See Figure 1)

$\displaystyle(x,y,z)=\frac{(x^{*},y^{*},z^{*})}{D_{j}},$ (1)

---
(a)
(b) (c)
Figure 1: (a) The geometry of the nozzle and the plate; (b,c) two examples of
trailing edge deformations of different wavenumbers.

The fluid velocity values are scaled by the jet velocity, $V_{j}$.

$\displaystyle(u,v,w)=\frac{(u^{*},v^{*},w^{*})}{V_{j}},$ (2)

Pressure values are scaled by the dynamic pressure, $\rho_{\infty}V_{j}^{2}$,
and temperature values by the freestream temperature, $T_{\infty}$. Reynolds
number, Mach number, and Prandtl number are defined as:

$\displaystyle Re=\frac{\rho_{\infty}V_{j}D_{j}}{\mu_{\infty}},\hskip
14.22636ptMa=\frac{V_{j}}{a_{\infty}^{*}},\hskip
14.22636ptPr=\frac{\mu_{\infty}C_{p}}{k_{\infty}}$ (3)

where $\mu_{\infty}$, $a_{\infty}$, and $k_{\infty}$ are the freestream
dynamic viscosity, speed of sound, and thermal conductivity, respectively.
$C_{p}$ is the specific heat at constant pressure. For all simulations, these
values are initialized for air as an ideal gas.

###  2.2 Governing Equations

The filtered Navier–Stokes equations can be expressed in conservative form as:

$\displaystyle\mathbf{Q}_{t}+\mathbf{F}_{\xi}+\mathbf{G}_{\eta}+\mathbf{H}_{\zeta}=\mathbf{S}.$
(4)

Here, the subscripts denote partial derivatives with respect to time and the
curvilinear coordinates. $\mathbf{Q}$ is the vector of conserved quantities:

$\mathbf{Q}=\frac{1}{J}\left\\{\rho,\;\rho\vec{u},\;E\right\\}^{T},$ (5)

$\rho=\rho^{*}/\rho_{\infty}$ is the non-dimensional density of the fluid,
$\vec{u}=(u,v,w)$ is the non-dimensional velocity vector in physical space,
and $E$ is the total energy. The flux vectors, $\mathbf{F}$, $\mathbf{G}$ and
$\mathbf{H}$, are given by:

$\displaystyle\mathbf{F}=\frac{1}{J}\left\\{\begin{array}[]{c}\rho U\\\
\rho\vec{u}U+\xi_{x_{i}}(p+\tau_{i1})\\\
EU+p\tilde{U}+\xi_{x_{i}}\Theta_{i}\end{array}\right\\},$ (9)
$\displaystyle\mathbf{G}=\frac{1}{J}\left\\{\begin{array}[]{c}\rho V\\\
\rho\vec{u}V+\eta_{x_{i}}(p+\tau_{i2})\\\
EV+p\tilde{V}+\eta_{x_{i}}\Theta_{i}\end{array}\right\\},$ (13)
$\displaystyle\mathbf{H}=\frac{1}{J}\left\\{\begin{array}[]{c}\rho W\\\
\rho\vec{u}W+\zeta_{x_{i}}(p+\tau_{i3})\\\
EW+p\tilde{W}+\zeta_{x_{i}}\Theta_{i}\end{array}\right\\},$ (17)

Here, the repeated indices indicate Einstein summation convention over
$i\in{1,2,3}$. $U,V,W$ are the contravariant velocity components,

$\displaystyle U=\xi_{x}u+\xi_{y}v+\xi_{z}w,$ (18) $\displaystyle
V=\eta_{x}u+\eta_{y}v+\eta_{z}w,$ (19) $\displaystyle
W=\zeta_{x}u+\zeta_{y}v+\zeta_{z}w,$ (20)

$\tau$ is the shear stress tensor,

$\tau_{ij}=\frac{\mu}{Re}\left[\left(\frac{\partial\xi_{k}}{\partial
x_{j}}\frac{\partial u_{i}}{\partial\xi_{k}}+\frac{\partial\xi_{k}}{\partial
x_{i}}\frac{\partial
u_{j}}{\partial\xi_{k}}\right)-\frac{2}{3}\delta_{ij}\frac{\partial\xi_{l}}{\partial
x_{k}}\frac{\partial u_{k}}{\partial\xi_{l}}\right],$ (21)

and $\Theta$ is the heat flux,

$\Theta_{i}=u_{j}\tau_{ij}+\frac{\mu}{(\gamma-1)M_{\infty}^{2}RePr}\frac{\partial\xi_{l}}{\partial
x_{i}}\frac{\partial T}{\partial\xi_{l}}$ (22)

Again, repeated indices (that are not present on the left-hand side) indicate
summation. The Jacobian of the curvilinear transformation from the physical
space to computational space is denoted by $J$, and $\mathbf{S}$ is a vector
of prescribed source terms. Pressure, temperature, and density are related by
the ideal gas equation of state:

$p=\frac{\rho T}{\gamma M_{\infty}^{2}}$ (23)

The dynamic viscosity and thermal conductivity $k$ are related to temperature
using the Sutherland’s equations in dimensionless form:

$\displaystyle\mu=T^{3/2}\frac{1+C_{1}/T_{\infty}}{T+C_{1}/T_{\infty}};\hskip
11.38109ptk=T^{3/2}\frac{1+C_{2}/T_{\infty}}{T+C_{2}/T_{\infty}},$ (24)

For air at sea level, $C_{1}=110.4$ K, and $C_{2}=194$ K.

There are no explicit subgrid scale turbulence terms in Equation (4) (see
Grinstein et al. [29]). Instead, the compressible Navier–Stokes equations are
solved within the framework of an implicit large eddy simulation, where
numerical filtering is applied to account for the missing sub-grid scale
energy. The numerical solver uses high-order finite difference approximations
for the spatial derivatives and explicit time marching. The time integration
is performed using a second order Adams–Bashforth method [30]:

$\mathbf{Q}^{n+1}=\mathbf{Q}^{n}+k\left[\sum_{\nu=0}^{K}\beta_{\nu}L(\mathbf{Q}^{n-\nu})\right]$
(25)

Here, the constants $\beta_{\nu}$ are chosen to give either the maximum order
of accuracy [30] or the lowest dispersion and dissipation. $L(\mathbf{Q})$ is
the residual.

The spatial derivatives are discretized using the dispersion-relation-
preserving schemes of Tam and Webb [31] or a high-resolution 9-point
dispersion-relation-preserving optimized scheme of Bogey and Bailly [32]. To
damp out the unwanted high wavenumber waves from the solution, high-order
spatial filters, as developed by Kennedy and Carpenter [33], are used. No slip
boundary conditions for velocity and adiabatic conditions for temperature are
imposed at the solid surfaces. Sponge layers are imposed near the far-field
boundaries in regions that are outside the flow domain of interest. The sponge
layers combined with grid stretching act to damp-out unwanted waves returning
from the farfield boundaries. For the LES simulations, the flow in proximity
to the wall is modeled using the Werner–Wengler model (Werner and Wengle
[34]), so some small turbulent flow structures may not captured adequately. At
the inflow boundary, a constant mean flow in the core region is combined with
a hyperbolic tangent function in the shear region, and disturbances in the
form of a superposition of Fourier modes with random amplitudes, frequencies,
and wavenumbers. These disturbances introduce three-dimensional flow
variations into the jet flow.

The mean flows used to initialize the LES are obtained from RANS simulations,
where a classical SST $k-\omega$ turbulence model (Menter [28]) is applied to
account for missing fluctuations. This model was found to perform well for jet
applications (see Mihaescu et al. [35])

In the present work, high-order, central-difference schemes are used to
achieve increased resolution of the propagating disturbances. A shock-
capturing technique suitable for simulations involving central differences in
space is required to avoid unwanted oscillations that may propagate from
discontinuities arising in supersonic flows. Shock capturing techniques are
employed based on the general explicit filtering framework, a straightforward
approach which introduces sufficient numerical viscosity in the area of the
discontinuities, and negligible artificial viscosity in the rest of the
domain. The technique of Bogey et al. [36] introduces selective filtering at
each grid vertex to minimize numerical oscillations, and shock-capturing in
the areas where discontinuities are present. This method has been proven to
work efficiently for high-order accurate, nonlinear computations.

###  2.3 Ffowcs-Williams Hawkings Acoustic Analogy Method

The Ffowcs-Williams and Hawkings (FW-H) equation [37, 38] is an inhomogeneous
wave equation that can be derived by manipulating the continuity equation and
the Navier–Stokes equations for a compressible fluid. The FW–H equation can be
written as:

$\displaystyle\frac{1}{a_{0}^{2}}\frac{\partial^{2}p^{\prime}}{\partial
t^{2}}-\nabla^{2}p^{\prime}=\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}\\{T_{ij}H(f)\\}$ (26) $\displaystyle-$
$\displaystyle\frac{\partial}{\partial x_{i}}\\{\left[P_{ij}n_{j}+\rho
u_{i}(u_{n}-v_{n})\right]\delta(f)\\}$ $\displaystyle+$
$\displaystyle\frac{\partial}{\partial
t}\\{\left[\rho_{0}v_{n}+\rho(u_{n}-v_{n})\right]\delta(f)\\}$

where $u_{i}$ is the fluid velocity component in the $x_{i}$ direction,
$u_{n}$ is the fluid velocity component normal to the surface at $f=0$,
$v_{i}$ is the surface velocity components in the $x_{i}$ direction, $v_{n}$
is the surface velocity component normal to the surface, $\delta(f)$ is Dirac
delta function, $H(f)$is Heaviside function, and $p^{\prime}=p-p_{0}$ is the
magnitude of the sound pressure disturbance at the far field.

The acoustic pressure in the farfield at a location, $\vec{\mathbf{x}}$, can
be determined from:

$p^{\prime}(\vec{\mathbf{x}},t)=p^{\prime}_{T}(\vec{\mathbf{x}},t)+p^{\prime}_{L}(\vec{\mathbf{x}},t)+p^{\prime}_{Q}(\vec{\mathbf{x}},t)$
(27)

where

$p^{\prime}_{T}(\vec{\mathbf{x}},t)=\frac{1}{4\pi}\frac{\partial}{\partial
t}\int_{S}\left[\frac{\rho_{0}U_{n}}{r}\right]_{ret}dS$ (28)
$p^{\prime}_{L}(\vec{\mathbf{x}},t)=\frac{1}{4\pi
a_{0}}\frac{\partial}{\partial
t}\int_{S}\left[\frac{L_{r}}{r}\right]_{ret}dS+\frac{1}{4\pi}\int_{S}\left[\frac{L_{r}}{r^{2}}\right]_{ret}dS$
(29)

Given a point, $\vec{\mathbf{y}}$, in the near-field acoustic source on
surface $S$, $r=|\vec{r}|=|\vec{\mathbf{x}}-\vec{\mathbf{y}}|$ is the distance
from the surface source location to the farfield position $\vec{\mathbf{x}}$;
$U_{n}$ is the scalar product of $\vec{U}=\rho\vec{u}/\rho_{0}$ and the unit
normal to the surface $S$ at $\vec{\mathbf{y}}$; $L_{r}$ is the scalar product
of $L_{i}=P_{ij}n_{i}+\rho u_{i}u_{n}$ and the unit vector in the direction
from $\vec{\mathbf{y}}$ to $\vec{\mathbf{x}}$;
$P_{ij}=-\tau_{ij}+p\delta_{ij}$ ; $\tau_{ij}$ is the viscous stress; $\rho$
is the instantaneous density; and $\rho_{0}$ and $a_{0}$ are the ambient
density and sound speed, respectively.

In Equation (27), $p^{\prime}_{Q}(\vec{\mathbf{x}},t)$ is the contribution
from the quadrupole, but it is neglected here. The integrals in (28) and (29)
are calculated at the emission time $\tau=t-r/a$, where $a$ is the speed of
sound and $t$ is the reception time. The flow variables at $\tau$ must be
interpolated from the flow data at multiple time iterations, $t$, due to its
dependency on $\vec{\mathbf{y}}$, which varies over the surface $S$
surrounding the jet.

For the numerical implementation of the FW-H method, a conical surface that
includes the entire jet is considered. The primitive variables and viscous
stresses are interpolated from the flow domain to the surface at every $b$
time steps (the value of $b$ depends on the desired time resolution) and
written out to a large number of files for post-processing. Given a reception
time $t$, the emission time $\tau$ is calculated for every small surface
element $\Delta S$ on the conical surface. Within a specific surface element
$\Delta S$ on the surface, an interpolation is necessary to obtain the flow
data at the emission time $\tau$, which is needed to calculate the integrals
in (28) and (29). Next, we determine the time derivatives of $U_{n}$ and
$L_{r}$ and evaluate the integrals in Equations (28) and (29). The acoustic
pressure then is calculated from Equation (27).

##  3 Results and Discussion

Results are reported and discussed in this section for a high aspect ratio
rectangular jet exhausting over a flat surface located underneath the jet
flow. Because the aspect ratio of the nozzle exit is high, we simplify the
problem by considering only a portion from the nozzle along the spanwise
direction. This simplification is based on the assumption that the flow is
statistically two-dimensional (an assumption also considered by [24]). Thus,
the spanwise length of the flow domain is set to be four times the height of
the nozzle, and periodic boundary conditions are imposed in the lateral
direction.

The layout of the configuration is illustrated schematically in Figure 1a. The
flat plate is oriented parallel to the jet axis and located beneath the jet
exit. The vertical separation of the plate from the jet axis ($d$) and the
distance from the jet exit to the trailing edge of the plate ($L$) were both
varied in this study. These distances are specified below and in Table 1 in
terms of multiples of the nozzle height ($D_{j}$). The vertical separation
considered were 0.65, 0.75, and 1 times the nozzle height, while the
horizontal distances to the trailing edge considered were 4, 5, and 6 times
the nozzle height. A 3D view of the geometry of the nozzle and the plate with
deformed trailing edge are shown in Figure 1. The deformations in the spanwise
direction are prescribed as a sine function while the amplitude is modulated
by a hyperbolic tangent function in the streamwise direction. Two combinations
of wavenumbers and amplitudes are shown in Figure 1b,c.

The Reynolds number, based on the jet velocity and the height of the nozzle,
is 500,000, and the acoustic Mach number is $0.8$. The amplitude of velocity
disturbances imposed at the nozzle exit are restricted to less than 3% of the
jet velocity.

Table 1: Position of the plate with respect to the nozzle (see also Figure 1). Case | Vertical Stand-Off Dist. (d) | Streamwise Dist. from the Nozzle Exit (L)
---|---|---
1 | 0.75 $D_{j}$ | 4 $D_{j}$
2 | 0.75 $D_{j}$ | 5 $D_{j}$
3 | 0.75 $D_{j}$ | 6 $D_{j}$
4 | 1.0 $D_{j}$ | 5 $D_{j}$
5 | 0.65 $D_{j}$ | 5 $D_{j}$

###  3.1 RANS Results

The mean flow and turbulent kinetic energy (TKE) used to initialize the LES
are obtained from separate two-dimensional RANS simulations using a
$\kappa-\omega$ turbulence model. The RANS results shown in Figures 2–6 are
reported first to demonstrate the effect of the plate location on the jet mean
flow and TKE. Plots of mean velocity magnitude (left) and turbulent kinetic
energy (right) are shown in Figure 2 for two selected cases, corresponding to
the largest and the smallest offset distance of the plate from the jet axis.
These color plots indicate that there is a visible effect from the plate on
both the mean flow and the turbulent kinetic energy. For the first
configuration ($d=1.0D_{j}$, $L=5D_{j}$), which corresponds to the largest
distance $d$ from the center of the nozzle, the jet is significantly deviated
downward by the plate (resembling the ’Coanda’ effect). This deviation is also
present in other cases, but they do not appear to be as significant as it is
for this case. The smallest deviation from the jet axis occurs for the
configuration ($d=0.65D_{j}$, $L=5D_{j}$), which corresponds to the smallest
offset distance, $d$.

From the plots of TKE shown on the right side of Figure 2, there appears to be
an attenuation of the TKE in the bottom shear layer of the jet as a result of
the interaction with the plate boundary layer. This is potentially due to the
viscous dissipation in the boundary layer that is developing on the surface of
the plate. Further downstream of the plate trailing edge, the magnitude of the
TKE appears to recover somewhat as the two shear layers show similar
intensities. This is further revealed by the profiles of TKE across the jet
shown in Figure 6.

---
(a) (b)
(c) (d)
Figure 2: Mean velocity magnitude (left) and turbulent kinetic energy (right)
distributions from RANS simulations of two cases. Top (a,b): $d=1.0D_{j}$,
$L=5D_{j}$; Bottom (c,d): $d=0.65D_{j}$, $L=5D_{j}$.

Figure 3: Mean streamwise velocity along the jet centerline from RANS: (a) $x$
from $0$ to $45$; (b) details in the core region with $x$ from $0$ to $12$.

Figure 4: Turbulent kinetic energy along the jet centerline from RANS: (a) $x$
from $0$ to $45$; (b) details in the core region with $x$ from $0$ to $12$.

Center-line mean velocity distributions are plotted in Figure 3 for several
tested configurations along with the results obtained from the baseline free
jet (no plate). The presence of the plate seems to slow down the flow inside
the potential core as seen in the zoomed-in plot shown in Figure 3b. While all
curves seem to follow the same trend in the downstream region, there is a
slight acceleration for the cases that include the plate. This does seem to
depend on the distance of the jet axis to the plate, $d$; the smaller this
distance, the higher the acceleration. TKE distributions along the jet center
line that are plotted in Figure 4 reveal a decrease of TKE in the downstream
region for the cases that involve the plate. This decrease is more significant
for the configurations with the smallest $d$ and smallest $L$ (yellow and red
curves in Figure 4).

In Figures 5 and 6, we compare cross-flow vertical profiles of mean velocity
magnitude and turbulent kinetic energy among the five configurations, at four
axial locations, $x=4D_{j}$;, $x=10D_{j}$, $x=25D_{j}$, and $x=45D_{j}$; the
results from the free jet case are also included. They all show that the
largest deviation of the jet from the original position occurs for
$d=1.0D_{j}$ and $L=5D_{j}$, corresponding to the largest distance of the
plate from the jet (this was also observed in the contour plots of the mean
velocity and TKE). Figure 6 shows that the reduction of the TKE in the lower
shear layer is more significant for the configuration with the smallest
distance from the trailing edge. Figures 5d and 6d indicate that the jet flow
becomes more symmetric further downstream for all normalized profiles, except
the case corresponding to $d=1.0D_{j}$, $L=5D_{j}$.

|
---|---
|

Figure 5: Streamwise velocity profiles along vertical direction, for different
configurations: (a) $x\leavevmode\nobreak\ =\leavevmode\nobreak\ 4D_{j}$; (b)
$x=10D_{j}$; (c) $x=25D_{j}$; (d) $x=45D_{j}$.

Figure 6: Turbulent kinetic energy profiles along vertical direction, for
different configurations: (a) $x=4D_{j}$; (b) $x=10D_{j}$; (c) $x=25D_{j}$;
(d) $x=45D_{j}$.

###  3.2 LES Results

A suite of LES runs were carried out corresponding to the geometrical
configurations given in Table 1, and the trailing edge deformations
characterized by the function

$\displaystyle
g(x,z)=0.5A((1+\tanh(\sigma_{x}(x+x_{0})))*\sin(2\pi(wz)z/L_{z})$ (30)

where $x_{0}=-1.2$ is the streamwise location on the plate where the
deformation begins to gradually increase from zero to the maximum amplitude at
the trailing edge location; $\sigma_{x}=2$ is a parameter controlling the
’thickness’ of the tangent hyperbolic function; and $L_{z}$ is the spanwise
width of the plate. The amplitude, $A$, and wavenumber, $k$, are varied as
given in Table 2. (See also Figure 1 for a visual representation of the
trailing edge deformations.) As mentioned previously, the spanwise length of
the flow domain is four times the height of the nozzle, and periodic boundary
conditions are imposed in the lateral direction with the assumption that the
aspect ratio of the rectangular nozzle is very high. This lateral dimension of
$4D_{j}$ was selected to ensure that all relevant turbulent flow structures
are captured accurately in the spanwise direction. The mesh consists of
approximately $10$ millions grid points, clustered mostly in proximity to the
jet region. Grid stretching is used in the farfield along with an imposed
sponge layer condition at the outflow boundary to ensure that the fluctuations
are gradually dissipated while they leave the flow domain.

Figure 7 shows iso-surfaces of Q-criterion colored by the streamwise velocity
component. ($Q=1/2[|\mathbf{\Omega}|^{2}-|\mathbf{S}|^{2}]$, where
$\mathbf{S}=1/2[\nabla\mathbf{v}+(\nabla\mathbf{v})^{T}]$ is the rate-of-
strain tensor, and
$\mathbf{\Omega}=1/2[\nabla\mathbf{v}-(\nabla\mathbf{v})^{T}]$ is the
vorticity tensor.) Acoustic waves radiating from the jet are plotted in gray
contours. (The white lines are caused by gaps between blocks from the MPI
decomposition). The attenuation of acoustic waves propagating from the jet
underneath the plate can be observed. This is a result of the shielding
provided by the plate causing the waves to not be as intense in this region.

Table 2: Amplitudes and wavenumbers of the trailing edge. Treatment |  Amplitude, A |  Wavenumber, wz
---|---|---
1 | 0.04 $D_{j}$ | 10
2 | 0.06 $D_{j}$ | 8
3 | 0.08 $D_{j}$ | 6
4 | 0.12 $D_{j}$ | 5
5 | 0.14 $D_{j}$ | 4
6 | 0.16 $D_{j}$ | 3
Figure 7: Iso-surfaces of Q-criterion colored by the streamwise velocity and
pressure contours in gray.

The mean centerline velocity distributions are compared against RANS results
in Figure 8. While there are some slight differences between the two in the
potential core region, overall the agreement is good, showing similar decay in
the downstream flow. (The LES profiles show some fluctuations from
insufficient time spans used for the time averaging).

|  |
---|---|---
|  |

Figure 8: Mean streamwise velocity along the jet centerline: (a)
$d=0.65D_{j}$, $L=5D_{j}$; (b) $d=0.75D_{j}$, $L=4D_{j}$; (c) $d=0.75D_{j}$,
$L=5D_{j}$; (d) $d=0.75D_{j}$, $L=6D_{j}$; (e) $d=1.0D_{j}$, $L=5D_{j}$; (f)
no plate.

Acoustic spectra have been calculated at three probe locations relatively
close to the jet (in the nearfield). These probes are located at large angles
with respect to the jet axis since it was observed that jet surface
interaction noise dominates at large polar angles (see [39] for a study of AR8
nozzle and [20] where various rectangular nozzle jets were considered). The
Ffowcs-Williams Hawkins surface is shown in Figure 9. The vertical side in the
downstream was placed at 30 nozzle diameters from the nozzle exit plane.
(Varying this distance did not affect the results significantly.) Two probes
are located above the jet while the third is located underneath the jet (see
Figure 9). A fourth probe was placed in the far field at 100 equivalent
diameters and $50$ deg angle with respect to the jet axis to compare the
numerical results with experimental results collected by Bridges et al. [40]
(see also Bridges [41]). Results were analyzed by using the sound pressure
level and overall sound pressure level plots. The Strouhal number, $St_{e}$,
was computed by the formula:

$St_{e}=\frac{fD_{j}}{V_{j}}$ (31)

where f is the frequency, $D_{j}$ is nozzle height, and $V_{j}$ is the jet
velocity.

Figure 9: Locations of the three Probes, and the Ffowcs-Williams integration
surface.

Figure 10 shows the comparison between our acoustic spectrum and the spectrum
obtained from measurements (Bridges et al. [40]). Bridges et al. [40]
conducted an extensive set of measurements to acquire acoustic data on single-
flow convergent rectangular nozzles of aspect ratios 2:1, 4:1, and 8:1, at
Mach numbers 0.7 and 0.9. Our comparison is for the largest aspect ratio 8:1
and Mach number 0.9. The agreement in Figure 10 is fairly good in the low
frequency range of the spectrum. For the high frequency range the spectrum
curve obtained from numerical simulations is slightly lower, which is most
likely due to the mesh resolution of the LES not being sufficient to fully
resolve the smallest flow scales that are responsible for the highest
frequency noise.

Table 3: Probe locations. Probe | X-Location | Y-Location
---|---|---
1 | 0 | 5
2 | 1 | 5.3
3 | 1 |  $-$3
Figure 10: Comparison between numerical results and measurements (Bridges et
al. [40]).

In general, the effect of trailing edge deformation on the radiated noise was
small. We noticed both an increase and a decrease in noise, depending on the
combination of amplitudes and wavenumbers characterizing the deformations. A
visible reduction in noise was achieved for treatment 5 given in Table 2
($w_{z}=4$, $A=0.14$). The acoustic spectra at the three probe points are
plotted for this case in Figure 11a–c.

|
---|---

Figure 11: Typical acoustic spectra that show a reduction: (a) Probe 1; (b)
Probe 2; (c) Probe 3.

The overall sound pressure levels (OASPL) observed at the probe points are
plotted in Figures 12 and 13. In Figure 12, the OASPL’s are plotted for the
five configurations that do not involve trailing edge deformations. For the
probes that are located above the jet, there is a significant increase in the
radiated noise for the case corresponding to the smallest distance from the
nozzle center ($d=0.65D_{j}$, $L=5D_{j}$). This is to be expected for this
case since the end of the potential core is the closest to the trailing edge
of the plate. For the probe located under the jet, there is an increase in the
radiated noise for the case corresponding to the shortest plate length. This
is an indication of a reduced shielding effect.

Figure 13 displays the OASPL results at all three probe locations for each of
the tested configurations and prescribed trailing edge deformations. Each plot
displays the results for all of the treatments given in Table 2 at one of the
fixed plate configurations given in Table 1. When the data are broken down by
each plate placement location, there were two apparent plate configurations
that yielded a noticeable reduction in sound levels at the probe locations. We
will look further into these configurations to determine if the trailing edge
deformations provide an additional positive effect.

|
---|---

Figure 12: Level of reduction of the overall sound pressure level for all
probes compared to no edge deformations: (a) Probe 1; (b) Probe 2; (c) Probe
3.

There appeared to be an overall reduction in noise for most of the treatments
in case 1 ($d=0.75D_{j}$ $L=4D_{j}$). The average overall sound pressure level
for the no deformation case was 121.367 dB. For the $w_{z}=10$ $A=0.04$
treatment, the average was 120.467 dB, and, for the $w_{z}=6$ $A=0.08$ case,
the average was 120.4 dB. The results for the Overall Sound Pressure Level
data can be seen in Table 4. Although there was a reduction in noise for all
of the deformations at this plate placement location, the two deformations
mentioned above demonstrated the greatest reduction in noise.

Table 4: Overall Sound Pressure Level results for the probes corresponding to the plate placement of $d$ = 0.75 $D_{j}$ $L$ = 4 $D_{j}$. Probe | No Def. | wz = 10 A = 0.04 | wz = 6 A = 0.08
---|---|---|---
1 | 121.5 | 120.7 | 120.8
2 | 122.1 | 121.0 | 121.6
3 | 120.5 | 119.7 | 118.8

|
---|---
|

Figure 13: Overall sound pressure level for trailing edge deformations: (a)
Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5.

For case 5 ($d=0.65D_{j}$ $L=5D_{j}$), there was also a reduction in noise for
all of the deformations applied to the trailing edge of the plate. The average
overall sound pressure level for the probes for the no deformation case was
122.067 dB. For the $w_{z}=10$ $A=0.04$ treatment, the average was 120.633 dB,
and for the $w_{z}=4$ $A=0.14$ treatment the average was 121.133 dB. The
Overall Sound Pressure Level results for this plate placement location can be
seen in Table 5.

Table 5: Overall Sound Pressure Level results for the probes corresponding to the plate placement of $d$ = 0.65 $D_{j}$ $L$ = 5 $D_{j}$. Probe | No Def. | wz = 10 A = 0.04 | wz = 4 A = 0.14
---|---|---|---
1 | 124.2 | 123.2 | 122.9
2 | 123.7 | 121.3 | 122.6
3 | 118.3 | 117.4 | 117.9

The OASPL results from these configurations yielded the best results for
overall noise reduction among those tested. It would be worthwhile to consider
other deformations that are close to these treatments as well as other nearby
plate placements to determine if further noise reduction could be achieved.

##  4 Conclusions

In this work, jet flow over various surface deformations at the trailing edge
of a plate installed under a high aspect ratio rectangular jet were simulated
to determine the effect of the jet-surface interaction on the noise radiated
to the farfield. RANS calculations and a suite of LES simulations were
performed with a high-order accurate flow solver. Numerous configurations and
flow conditions were investigated, and the emitted noise from the jet-surface
interactions calculated using Ffowcs-Williams and the Hawkings acoustic
analogy method.

The RANS results, consisting of centerline velocity distributions, showed that
the presence of the plate appeared to slow down the flow inside the potential
core. For some of the tested cases, a slight acceleration of the flow further
in the downstream was observed. The amount of acceleration appears to depend
on the offset distance of the plate from the center of the jet, $d$. TKE
distributions along the jet centerline revealed a decrease in TKE magnitude in
the downstream region, which was expected. This decrease was more significant
for the configuration with the smallest $d$ and smallest $L$. Cross-flow
profiles of mean velocity magnitude and turbulent kinetic energy at different
axial locations showed a clear deviation of the jet from the axis of symmetry
near the trailing edge. The TKE profiles indicated that the jet is slightly
skewed around the trailing edge of the plate in a manner resembling the Coanda
effect. Despite this deviation, the cross-flow profile becomes more symmetric
further downstream. A reduction of the TKE in the lower shear was observed for
all cases with the plate under the jet. This is most significant for the
configurations with the smallest distance to the trailing edge.

For the LES analysis, various results consisting of iso-surface of
Q-criterion, centerline velocity, acoustic spectra, and overall sound pressure
levels were presented and discussed. Overall, good agreement was found between
the centerline velocities from RANS and LES, with some small discrepancies in
the potential core region. Acoustic spectra plotted for the configurations
without trailing edge deformations confirmed a reduction of the noise under
the plate from the shielding effect. For most cases involving the trailing
edge deformations, there did not appear to be a significant change to overall
sound levels. However, some cases did demonstrate a noticeable increase or
decrease in emitted noise. In the best cases, the overall sound pressure
levels were decreased from 1 to 5 dB.

The results presented in this work provide evidence that further refinement of
the configurations described herein may yield additional noise reductions.
Future research aimed at analyzing the interactions of the jet flow with
different types of surface deformations (e.g., non-periodic and/or streamwise-
oriented) may uncover a greater degree of noise reduction, beyond what our
results have shown.

## References

  * [1] Brown, C.A.; Wernet, M.P. Jet surface interaction test: Flow measurements results. In Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, USA, 16–20 June 2014; AIAA Paper 2014-3198.
  * [2] Brown, C.A. Jet-Surface Interaction Test: Far-Field Noise Results. J. Eng. Gas Turbines Power 2013, 135, 071201.
  * [3] Podboy, G. Jet-Surface Interaction Test: Phased Array Noise Source Localization Results; ASME GT2012-69801; American Society of Mechanical Engineers: New York, NY, USA, 2012.
  * [4] Bridges, J.; Wernet, M.P. The NASA Subsonic Jet ParticleImage Velocimetry (PIV) Dataset; NASA TM 2011-216807; NASA: Washington, DC, USA, 2011.
  * [5] Brown, C.; Bridges, J. Small Hot Jet Acoustic Rig Validation; NASA TM 2006-214234; NASA: Washington, DC, USA, 2006.
  * [6] Gutmark, E.J.; Grinstein, F.F. Flow control with noncircular jets. Annu. Rev. Fluid Mech. 1999, 31, 239–272.
  * [7] Gao, N.; Sun, H.; Ewing, D. Heat transfer to impinging round jets with triangular tabs. Int. J. Heat Mass Transf. 2003, 46, 2557–2569.
  * [8] El Hassan, M.; Meslem, A.; Abed-Meraim, K. Experimental investigation of the flow in the near-field of a cross-shaped orifice jet. Phys. Fluids 2011, 23, 045101.
  * [9] Violato, D.; Ianiro, A.; Cardone, G.; Scarano, F. Three-dimensional vortex dynamics and convective heat transfer in circular and chevron impinging jets. Int. J. Heat Fluid Flow 2012, 37, 22–36.
  * [10] Shakouchi, T.; Iryama, S. Flow characteristics of submerged free jet flow from petal-shaped nozzles. J. Fluid Sci. Technol. 2014, 9, 1–7.
  * [11] Malla, B.; Gutmark, J. Single Expansion Ramp Nozzles: Impact of Ramp Length on Flow and Acoustics. AIAA J. 2021, 59, 3152–3164.
  * [12] Mancinelli, M.; Di Marco, A.; Camussi, R. Multivariate and conditioned statistics of velocity and wall pressure fluctuations induced by a jet interacting with a flat plate. J. Fluid Mech. 2017, 823, 134–165, doi:10.1017/jfm.2017.307
  * [13] Proenca, A.; Lawrence, J.; Self, R. Investigation into the turbulence statistics of installed jets using hot-wire anemometry. Exp. Fluids 2020, 61, 220, https://doi.org/10.1007/s00348-020-03054-x.
  * [14] Mohan, N.K.D.; Doty, M.J. Active Chevrons For Jet Noise Reduction. In Proceedings of the 24th International Congress On Sound In addition, Vibration, London, UK, 23–27 July 2017.
  * [15] Butler, G.; Calkins, F.T. Initial attempts to suppress jet noise using piezoelectric actuators. In Proceedings of the 9th AIAA/CEAS Aeroacoustics Conference, Hilton Head, SC, USA, 12–14 May 2003; No. 2003-3192.
  * [16] Zaman, K.B.M.Q. Jet Noise Reduction by Microjets—A Parametric Study; AIAA-2009-3129; NASA/TM-2010-215846; NASA Glenn Research Center: Cleveland, OH, USA, 2010.
  * [17] Semlitsch, B.; Cuppoletti, D.; Gutmark, E.; Mihaescu, M. Transforming the Shock Pattern of Supersonic Jets Using Fluidic Injection. AIAA J. 2019, 57, 1851–1861.
  * [18] Rego, L.; Avallone, F.; Ragni, D.; Casalino, D. Noise Amplification Effects due to Jet-Surface Interaction. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019; AIAA 2019-0001.
  * [19] Brown, C. Jet-Surface Interaction Test: Far-Field Noise Results. In Proceedings of the ASME Turbo Expo 2012: Power for Land, Sea and Air, Copenhagen, Denmark, 11–15 June 2012; pp. 1-13.
  * [20] Afsar, M.; Leib, S.; Bozak, R. Effect of de-correlating turbulence on the low frequency decay of jet-surface interaction noise in sub-sonic unheated air jets using a CFD-based approach. J. Sound Vib. 2017, 386, 177–207.
  * [21] Behrouzi, P.; McGuirk, J.J. Underexpanded Jet Development from a Rectangular Nozzle with Aft-deck. AIAA J. 2015, 53, 1287–1298.
  * [22] Bridges, J.E. Noise from Aft Deck Exhaust Nozzle Differences in Experimental Embodiments. In Proceedings of the 52nd AIAA Aerospace Meeting, National Harbor, MD, USA, 13–17 January 2014; AIAA 2014-0876.
  * [23] Zaman, K.B.M.Q.; Fagan, A.F.; Clem, M.M.; Brown, C.A. Resonant Interaction of a Rectangular Jet with a Flat-plate. In Proceedings of the 52nd AIAA Aerospace Meeting, National Harbor, MD, USA, 13–17 January 2014; AIAA 2014-0877.
  * [24] Zaman, K.B.M.Q.; Brown, C.A.; Bridges, J.E. Interaction of a Rectangular Jet with a Flat-Plate Placed Parallel to the Flow; AIAA Paper 2013-2184; NASA/TM-2013-217879 (E-18684); AIAA: Reston, VA, USA, 2013.
  * [25] Seiner, J.M.; Manning, J.C. Supersonic Jet Plume Interaction with a Flat Plate. In Proceedings of the International Power Lift Conference, Santa Clara, CA, USA, 17–19 March 1987.
  * [26] Ibrahim, M.K.; Sawai, T.; Obase, K.; Mori, K.; Nakamura, Y. Experimental Investigation of Screech Tone Characteristics of Jet Interaction with a Flat Plate. AIAA J. 2009, 47, 2031–2038.
  * [27] Berland, J.; Bogey, C.; Bailly, C. Numerical study of screech generation in a planar supersonic jet, Physics of Fluids. Am. Inst. Phys. 2007, 19, 075105.
  * [28] Menter, F.R. Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA J. 1994, 32, 1598–1605.
  * [29] Grinstein, F.F.; Margolin, L.G.; Rider, W.J. Implicit Large Eddy Simulation; Cambridge University Press: Cambridge, UK, 2007.
  * [30] Butcher, J.C. Numerical Methods for Ordinary Differential Equations; John Wiley: Hoboken, NJ, USA, 2003.
  * [31] Tam, C.K.W.; Webb, J.C. Dispersion-relation-preserving finite difference schemes for Computational Aeroacoustics. J. Comput. Phys. 1993, 107, 262–281.
  * [32] Bogey, C. ; Bailly, C. A family of low dispersive and low dissipative explicit schemes for flow and noise computation. J. Comput. Phys. 2004, 194, 194–214.
  * [33] Kennedy, C.A.; Carpenter, M.H. Comparison of Several Numerical Methods for Simulation of Compressible Shear Layers; NASA Technical Report NASA-97-TP3484; NASA: Washington, DC, USA 1997.
  * [34] Werner, H.; Wengle, H. Large-eddy simulation of turbulent flow over and around a cube in a plate channel. In Turbulent Shear Flows 8; Springer: Munich, Germany, 1993; Volume 8.
  * [35] Mihaescu, M.; Semlitsch, B.; Fuchs, L.; Gutmark, E. Assessment of turbulence models for predicting coaxial jets relevant to turbofan engines. In Proceedings of the Conference on Modelling Fluid Flow (CMFF 12), The 15th International Conference on Fluid Flow Technologies, Budapest, Hungary, 4–7 September 2012.
  * [36] Bogey, C.; Bailly, C.; Juve, D. A shock-capturing methodology based on adaptative spatial filtering for high-order nonlinear computations. J. Comput. Phys. 2002, 228, 1447–1465.
  * [37] Brentner, K.S.; Farassat, F. An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulations for Moving Surfaces. AIAA J. 1998, 36, 1379–1386.
  * [38] Ffowcs-Williams, J.E.; Hawkings, D.L. Sound Generation by Turbulence and Surfaces in Arbitrary Motion. Philos. Trans. R. Soc. Lond. 1969, A264, 321–342.
  * [39] Goldstein, M.E.; Leib, S.J.; Afsar, M.Z. Generalized rapid-distortion theory on transversely sheared mean flows with physically realizable upstream boundary conditions: application to trailing edge problem. J. Fluid Mech. 2017, 824, 477–512.
  * [40] Bridges, J.E.; Brown, C.A.; Bozak, R.F. Experiments on exhaust noise of tightly integrated propulsion systems. In Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, USA, 16–18 June 2014; AIAA Paper 2014-2904.
  * [41] Bridges, J.E. Noise from Aft Deck Exhaust Nozzles - Differences in experimental embodiments. In Proceedings of the 52nd Aerospace Sciences Meeting, National Harbor, MD, USA, 3–17 January 2014; AIAA Paper 2014-0876.
  * [42] Bres, G.; PÃ©rot, F.; Freed, D. Properties of the Lattice Boltzmann Method for Acoustics. In Proceedings of the 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), Miami, FL, USA, 11–13 May 2009, doi:10.2514/6.2009-3395.
  * [43] Chen, H.; Kandasamy, S.; Orszag, S.A.; Succi, S.; Yakhot, V. Extended Bolzmann Kinetic Equation for Turbulent Flows. Science 2003, 301, 633–636.
  * [44] Conrad, E.W.; Ciepluch, C.C. Jet Aircraft Engine Noise Reduction; NASA Technical Memorandum NASA TM X-68131; NASA: Washington, DC, USA, 1972.
  * [45] Fares, E.; Casalino, D.; Khorrami, M.R. Evaluation of Airframe Noise Reduction Concepts via Simulations using a Lattice-Boltzmann Approach. In Proceedings of the 21st AIAA/CEAS Aeroacoustics Conference , Dallas, TX, USA, 22–26 June 2015; AIAA Paper 2015-2988.
  * [46] Fares, E.; Duda, B.; Khorrami, M.R. Airframe Noise Prediction of a Full Aircraft in Model and Full Scale Using a Lattice Boltzmann Approach. Presented at the 22nd AIAA/CEAS Aeroacoustics Conference, Lyon, France, 30 May–1 June 2016.
  * [47] Khorrami, M.R.; Duda, B.; Hazir, A.; Fares, E. Computational Evaluation of Airframe Noise Reduction Concepts at Full Scale. In Proceedings of the 22nd AIAA/CEAS Aeroacoustics Conference, Lyon, France, 30 May–1 June 2016; AIAA 2016-2711.
  * [48] Khorrami, M.R.; Fares, E.; Casalino, D. Towards Full-Aircraft Airframe Noise Prediction: Lattice-Boltzmann Simulations. In Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, USA, 16–18 June 2014; AIAA Paper 2014-2481.
  * [49] Yakhot, V.; Orszag, S.A. Renormalization Group Analysis of Turbulence. I. Basic Theory. J. Sci. Comput. 1986, 1, 3–51.