# Physics-Informed Acoustic Liner Optimization:
Balancing Drag and Noise

Haris Shahzad  Aerodynamics Group, Faculty of Aerospace Engineering, Delft
University of Technology,
Kluyverweg 2, 2629 HS Delft, The Netherlands  Stefan Hickel and Davide Modesti
Aerodynamics Group, Faculty of Aerospace Engineering, Delft University of
Technology,
Kluyverweg 2, 2629 HS Delft, The Netherlands

###### Abstract

We present pore-resolved Direct Numerical Simulations (DNS) of turbulent flows
grazing over acoustic liners with aerodynamically and/or acoustically
optimized orifice configurations. Our DNS explore a large parameter space,
studying various families of orifice geometries, including the influence of
orifice shape, orientation, and the number of orifices. All flow cases show an
increase in drag compared to the smooth wall. However, the added drag can be
reduced by as much as $\sim$55% as compared to conventional acoustic liners by
simply altering the shape of the orifice or its orientation, in the case of a
non-circular orifice. Complementary acoustic simulations demonstrate that this
reduced drag may be achieved while maintaining the same noise reduction
properties over a wide range of frequencies.

$\omega$ = | angular frequency
---|---
$A$ = | cross-sectional orifice area
$V_{c}$ = | cavity volume
$V_{o}$ = | orifices’ volume
$h$ = | cavity depth
$t$ = | orifice thickness
$c$ = | speed of sound
$f$ = | frequency of sound wave
$f_{r}$ = | liner resonance frequency
$C_{f}$ = | friction coefficient
$d$ = | orifice diameter
$a$ = | major axis of ellipse
$b$ = | minor axis of ellipse
$h$ = | cavity depth
$L_{\chi}$ = | domain length in $\chi$ direction
$M_{\infty}$ = | Freestream Mach number
$M_{b}$ = | Bulk Mach number
$N_{\chi}$ = | grid points in $\chi$ direction
$Re$ = | Reynolds number
$u_{b}$ = | bulk velocity
$u_{\text{cl}}$ = | centerline velocity
$u_{\tau}$ = | friction velocity
$\alpha$ = | Forchheimer permeability
$K$ = | Darcy permeability
$L_{\chi}$ = | Domain size in $\chi$ direction
$\ell_{T}$ = | virtual origin
$\delta$ = | channel half width
$\delta_{v}$ = | viscous length scale
$\Delta U^{+}$ = | Hama roughness function
$\lambda_{c}$ = | cavity cross-section length/width
$\nu$ = | kinematic viscosity
$\rho$ = | density
$\sigma$ = | facesheet porosity
$\tau_{ij}$ = | Reynolds stresses
$\tau_{w}$ = | wall shear stress

##  1 Introduction

Aircraft engines are the primary source of noise during take-off and landing.
To reduce noise, engine nacelles are equipped with noise control devices
called acoustic liners. Acoustic liners are essentially an array of Helmholtz
resonators, which consist of a large cavity with a small opening in the top
plate. Acoustic liners are simple devices, and their working mechanism is
based on the idea of dissipating noise by tuning their resonance frequency
with the dominant frequency of the engine fan. They have the potential to
attenuate noise by as much as 8–10dB and are thus, required on all aircraft
jet engines.

Acoustic liners have been optimised primarily from an acoustic perspective [1,
2, 3]. From an aerodynamic perspective, however, they behave like surface
roughness and contribute to an increase in aerodynamic drag. Our recent study
[4] showed that acoustic liners are responsible for about a 70% drag increase
per plane area compared to a hydraulically smooth wall at typical operating
conditions. This aerodynamic penalty has been accepted as a necessary
compromise.

The effect of acoustic liners on the background turbulent flow has only been
studied recently, both using experiments [5, 6, 7, 8] and numerical
simulations [9, 10]. Experimental studies are often characterized by large
uncertainties in drag measurements, and numerical simulations of such surfaces
are computationally expensive. Therefore, most studies have used simplifying
assumptions to make the problem more approachable with numerical simulations.
Several authors simulated the flow over a single acoustic liner cavity [11,
12, 10] or replaced the acoustic liner with an equivalent boundary condition
[13, 14, 15]. As a result, estimates of the added drag reported in the
literature have a massive spread, ranging from 3% up to 500%, depending on the
numerical or experimental technique that was used [4].

Our group recently performed the first Direct Numerical Simulations (DNS) of
fully resolved acoustic liner geometries [4] and we related the added drag to
the wall-normal velocity fluctuations and the wall-normal Forchheimer
permeability of the plate. Building on these findings, we showed that it is
possible to predict the added drag of acoustic liners in operating conditions
[16, 4]. Hence, we are confident that better aerodynamic performance can be
achieved by finding plate geometries that reduce the wall-normal velocity
fluctuations induced by the orifices.

Howerton and Jones [17] studied different orifice configurations, changing the
orifice shape and orientation, including rectangular orifices either parallel
or perpendicular to the flow, and found that the perpendicular slot orifice
performed better compared to the baseline circular orifices. The parallel slot
orifice, on the other hand, despite having the same dimensions as the
perpendicular slot orifice, had the highest added drag (at a freestream Mach
number $M_{\infty}=0.3$). Furthermore, Howerton and Jones [17] noted that,
despite changes in the orifice shape, the acoustic performance was largely
unchanged, meaning that there is potential for reducing the added drag,
without hampering the acoustic attenuation properties.

To the best of our knowledge, acoustic liners with varying orifice
configurations have been studied only experimentally [18, 17] and the physical
rationale behind their choice of orifice configuration was missing. The
objective of the current study is to show that it is possible to reduce this
aerodynamic penalty while maintaining or improving acoustic noise attenuation.
In this work, we use our recent findings [4] on scaling laws for the added
drag to devise optimized acoustic liner geometries that reduce the added drag
while having the same acoustic performance as the baseline configuration.

##  2 Methodology

Figure 1: Sketch of the computational domain.

| $Re_{b}$ | $Re_{\tau}$ | $d_{x}^{+}$ | $d_{z}^{+}$ | $K/t^{2}\times 10^{3}$ | $1/(\alpha t)$ | $\Delta x^{+}$ | $\Delta y^{+}_{\text{min}}$ | $\Delta z^{+}$ | $N_{x}$ | $N_{y}$ | $N_{z}$ | $\Delta U^{+}$ | $C_{f}\times 10^{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$S$ | 18536 | 506.1 | 0 | 0 | - | - | 5.1 | 0.80 | 5.1 | 300 | 350 | 150 | - | 4.58
$L$-$S$ | 16528 | 505.3 | 40.4 | 40.4 | 6.33 | 0.127 | 1.5 | 0.80 | 1.5 | 1000 | 500 | 500 | 1.90 | 5.54
$L$-$T$ | 16984 | 515.8 | 33.0 | 33.0 | 17.0 | 0.156 | 1.3 | 0.81 | 1.3 | 1200 | 600 | 500 | 1.45 | 5.28
$L$-$E_{x-3}$ | 13124 | 492.8 | 29.6 | 157.7 | 7.32 | 0.0869 | 5.9 | 0.79 | 1.1 | 250 | 500 | 700 | 4.87 | 7.78
$L$-$E_{x-6}$ | 15602 | 501.5 | 15.0 | 160.5 | 2.23 | 0.0401 | 6.0 | 0.80 | 0.6 | 250 | 500 | 1300 | 2.33 | 5.80
$L$-$E_{x-9}$ | 17300 | 515.5 | 10.3 | 165.0 | 1.07 | 0.0217 | 6.2 | 0.82 | 0.4 | 250 | 500 | 2000 | 0.85 | 4.97
$L$-$E_{z-3}$ | 16104 | 510.0 | 163.2 | 30.6 | 7.32 | 0.0869 | 1.1 | 0.82 | 6.1 | 1400 | 500 | 125 | 2.17 | 5.70
$L$-$E_{z-6}$ | 16042 | 506.8 | 162.1 | 15.2 | 2.23 | 0.0401 | 0.6 | 0.81 | 6.1 | 2600 | 500 | 125 | 2.29 | 5.78
$L$-$E_{z-9}$ | 16670 | 515.6 | 165.0 | 10.3 | 1.07 | 0.0217 | 0.4 | 0.82 | 6.2 | 4000 | 500 | 125 | 1.90 | 5.54

Table 2: DNS dataset comprising smooth ($S$) and liner ($L$-$C$) cases.
Simulations are performed at bulk Reynolds number $Re_{b}=2u_{b}\delta/\nu$.
$d_{x}$ and $d_{z}$ are the lengths of the streamwise and spanwise axes of the
orifices. $\Delta x^{+}$, $\Delta y^{+}_{\text{min}}$ and $\Delta z^{+}$ are
the viscous-scaled streamwise, minimum wall-normal and spanwise mesh spacing.
Figure 2: Orifice configurations considered. All liner cases, except for case
$L$-$T$ have a top surface open area ratio of $\sigma=0.322$.

###  2.1 Test Setup

We solve the compressible Navier–Stokes equations for a calorically perfect
gas using the solver STREAmS [19, 20] in a plane channel flow configuration.
The computational domain is a rectangular box of size $L_{x}\times L_{y}\times
L_{z}=3\delta\times 2(\delta+h)\times 1.5\delta$, where $\delta$ is the
channel half-width, $h$ is the cavity depth, and $x,y,z$ denote the
streamwise, wall-normal and spanwise directions, respectively. This domain
size is chosen based on previous studies on wall roughness and acoustic liners
[4, 21, 22, 23, 24]. We perform DNS that fully resolves all turbulent length
and time scales using a sixth-order spatial discretization and a third-order,
three-stage Runge–Kutta scheme for time marching [25]. We carry out all
simulations at friction Reynolds numbers
$\mbox{{Re}}_{\tau}=\delta/\delta_{v}\approx 500$, where $\delta_{v}$ is the
viscous length scale. Liner flow cases are complemented by a smooth-wall
simulation at approximately the same friction Reynolds number and the same
domain size. The simulations are performed at bulk Mach number
$M_{b}=u_{b}/c_{w}=0.3$, where $u_{b}$ is the bulk flow velocity and $c_{w}$
is the speed of sound at the wall. A uniform body force $\Pi$ is added in the
streamwise direction and adjusted each time step to maintain a constant mass
flow rate in the channel core. Conservation of mean-momentum implies that the
body force is related to the total drag experienced by the solid wall as
$\Pi=\tau_{w}/\delta$, where $\tau_{w}$ is the drag per plane area. The
acoustic liner geometry is handled using a ghost-point forcing immersed
boundary method [26]. The immersed boundary method has been previously
verified and validated and has been used to study a wide range of geometries
[27, 4]. Both walls are covered by an array of $8\times 4$ acoustic liner
cavities, as illustrated in Fig. 1. Each cavity has a square cross-section
with a side length $\lambda_{c}=0.335\delta$ and depth $h=0.5\delta$. The
cavity walls have a thickness of $0.02\delta$. We use uniform mesh spacing in
the streamwise and spanwise directions. In the wall-normal direction, the mesh
is clustered towards the wall and coarsened towards the backplate and the
channel centre. A minimum of 26 mesh points are used to resolve the orifice
diameter (or minor axis in the case of an ellipse). This resolution is well
within the viscous spacing typically accepted in DNS, and it has been
previously validated [4].

We wish to study the influence of orifice geometry on the aerodynamic and
acoustic performance of a liner and compare it to the baseline configuration
with circular straight holes that are common in most applications. For the
baseline reference case, see Fig. 2 (b) we consider the geometry studied by
Shahzad et al. [4] with porosity $\sigma=0.322$, viscous-scaled diameter
$d^{+}=d/\delta_{v}\approx 40$ and a thickness to diameter ratio $t/d=1$,
where the orifice diameter for the baseline case is $d=0.08\delta$. We chose
the baseline liner geometry to be representative of acoustic liners in
operating conditions, while also keeping a reasonable computational cost to
perform a parametric study using DNS. The orifice size and facesheet thickness
are in the range found on engine-mounted acoustic liners when scaled with the
local boundary layer thickness in landing conditions, as discussed in Shahzad
et al. [4]. The cavity depth we use is smaller than in real applications,
however, previous studies have reported that its influence on the added drag
is negligible [5]. The porosity is higher than typically found on modern
aircraft engines. However, in our recent study [4] we demonstrate that the
relevant nondimensional parameter for quantifying the added drag is the
viscous-scaled Forchheimer permeability, which is thus a more relevant
nondimensional quantity than the porosity.

To compare the result of liner simulations with the smooth wall data, we take
into account the effect of the virtual origin $\ell_{T}$, namely, the distance
below the plate at which the flow perceives the equivalent smooth wall. More
details on how the virtual origin is estimated are provided in our previous
work [4]. In the following, quantities that are non-dimensionalised by
$\delta_{v}$ and $u_{\tau}$ are denoted by the ‘$+$’ superscript. The overline
symbol $\overline{f}$ is used to indicate Reynolds averages, whereas the tilde
$\widetilde{f}=\overline{\rho f}/\overline{\rho}$ indicates Favre averaging,
and the double prime symbol indicates fluctuations thereof
$f^{\prime\prime}=f-\widetilde{f}$, where $\rho$ is the density. As an
example, with this notation, the ensemble-averaged Reynolds stress tensor is
$\tau_{ij}=\overline{\rho}\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}$,
where $u_{i}=\\{u_{1},u_{2},u_{3}\\}=\\{u,v,w\\}$ are the streamwise, wall-
normal and spanwise velocity, respectively. All simulations are advanced
forward in time until they reach a statistically stationary state, after which
statistics are collected for at least $T_{\text{av}}u_{\tau}/\delta\approx
16$, where $T_{\text{av}}$ is the averaging interval.

###  2.2 Novel Configurations

The facesheet thickness and the cavity dimensions remain unchanged for all
cases considered. The novel configurations proposed only differ in the orifice
shape, size, and orientation. In an attempt to reduce the aerodynamic drag
induced by acoustic liners, we pursue two ideas:

  * •

We aim at increasing the wall-normal Forchheimer permeability of the plate
$\alpha_{y}$. Although $\alpha_{y}$ has a complex dependency on the plate
geometry, a first-order approximation is $\alpha_{y}\sim 1/(\sigma^{2}t)$
[16], and therefore a reduction of the plate permeability will result in lower
drag.

  * •

We take inspiration from riblets, namely streamwise-aligned surface grooves
that are able to reduce friction drag [28, 29], and argue that the same
surface anisotropy in the streamwise direction might be beneficial for
acoustic liners.

Following these two hypotheses, we propose liner geometries that should be
more efficient from an aerodynamic perspective and possibly retain the
acoustic properties of the liner. Therefore, we change the orifice geometry
while keeping constant the resonance frequency of the resonators
$\omega_{r}=c_{w}\sqrt{{A}/{tV_{c}}}$, where $A$ is the plane area of the
orifice, and $V_{c}$ is the volume of the cavity. In this way, we aim to
optimise the aerodynamic performance without compromising the acoustic
properties.

Based on the first idea, we propose a ‘tapered-hole’ configuration where the
orifice has a smaller diameter at the top of the facesheet and a larger
diameter at the bottom of the facesheet, such that the total volume of the
orifice is constant and the resonance frequency of the liner, disregarding
entry and exit effects, does not change. Entry and exit effects are included
using empirical correction factors when determining the resonance frequency of
the liner as they change the effective mass of air that oscillates in the
orifice. For a canonical orifice shape, the correction is known and well-
documented [30, 31, 32, 10]. Since entry/exit effects are accounted for by
empirical corrections, our hypothesis can only be verified a posteriori
through acoustic simulations at different frequencies. For the considered case
(case $L$-$T$) the orifice diameter increases continuously from
$d=0.064\delta$ at the top of the facesheet to $d=0.1024\delta$ at the bottom
of the facesheet.

Based on the second idea, we propose elliptical orifices, that have the same
porosity as the baseline liner, i.e. $\sigma=0.322$. The major axis of the
ellipse is fixed at $a=0.32\delta$ and the minor axis is calculated by
assuming a constant porosity and depends on the number of orifices per cavity.
For example, in the case of 9 elliptical orifices per cavity, see Fig. 2 (f)
or (i), the minor axis is $b=0.02\delta$, and in the case of 3 elliptical
orifices per cavity, see Fig. 2 (d) or (g), the minor is $b=0.06\delta$, such
that the porosity is always $\sigma=0.322$.

In addition, the influence of the elliptical orifices’ orientation is also
studied. While we expect that streamwise-aligned ellipses should minimize the
added drag, previous experiments [17] have shown that rectangular slots
perpendicular to the flow have lower drag compared to the canonical
configuration. Therefore, we consider elliptical orifices with the major axis
aligned with the streamwise or spanwise direction.

The geometries considered are shown in Fig. 2, and details of all flow cases
are reported in Table 2. The naming of the flow cases is as follows: we use
the letter $S$ for the smooth wall case and $L$-$C$ for the liner cases, where
$C=\\{S,T,E_{x-\chi},E_{z-\chi}\\}$ refers to the specific liner flow case. In
particular, $S$, $T$, $E_{x-\chi}$ and $E_{z-\chi}$ are the baseline liner
with straight orifice, tapered-orifice liner, streamwise-oriented ellipses and
spanwise-oriented ellipses, respectively, and $\chi$ is the number of ellipses
per cavity.

##  3 Results

###  3.1 Aerodynamic Drag

(a) |  (b)
---|---
Figure 3: Mean streamwise velocity for streamwise-oriented ellipses as a
function of $y^{+}$. Novel geometries are compared to the smooth wall flow
case (dashed black line without symbols) and the baseline liner flow case
(solid black line without symbols). The novel liner flow cases have the
following line style: $L$-$T$ (circles), $L$-$E_{x-\chi}$ (squares) and
$L$-$E_{z-\chi}$ (triangles). Different line types with symbols indicate the
number of orifices per cavity: $\chi=3$ (dash-dotted line with filled
symbols), $\chi=6$ (dashed line with half-filled symbols) and $\chi=9$ (solid
line with empty symbols).

Figure 3 shows the mean streamwise velocity profiles for all flow cases. We
see that the mean velocity profile over acoustic liners is shifted downward as
compared to the smooth wall. This downward shift is referred to as the $\Delta
U^{+}$, and it is directly related to the added drag through the exact
relation [4],

$\Delta D=1-\frac{C_{f}}{C_{f,s}}=1-\left(\frac{1}{1-\frac{\Delta
U^{+}}{\sqrt{2/C_{f,s}}}}\right)^{2},$ (1)

where $C_{f}=2\Pi\delta/(\rho U_{\text{c}}^{2})$ is the friction coefficient,
$U_{\text{c}}$ is the mean streamwise velocity at the channel centerline, and
the subscript ‘s’ denotes smooth wall values. The advantage of using $\Delta
U^{+}$, instead of the relative variation of the friction coefficient, is that
$\Delta U^{+}$ is independent from the Reynolds number, thus it can be used as
a drag measure in laboratory experiments or simulations which necessarily have
a lower Reynolds number than acoustic liners in operating conditions. The
effect of Reynolds number on $\Delta D$ is embedded in $C_{f,s}$, which can be
easily estimated using smooth-wall formulas, allowing one to use low Reynolds
number data to estimate the drag variation in operating conditions, as
discussed by Shahzad et al. [4]. A positive $\Delta U^{+}$ is correlated with
a drag increase and represents a downward shift of the velocity profile with
respect to the smooth wall. In Fig. 3 we note that all liner cases show a
downward shift compared to the smooth wall ($\Delta U^{+}$), indicating that
all cases increase drag. However, several liner geometries exhibit a lower
$\Delta U^{+}$ compared to the baseline case, demonstrating that modifying the
orifice shape can result in a lower added drag. For convenience, the value of
the Hama roughness function $\Delta U^{+}$ and the friction coefficient are
also reported in Table 2.

Some elliptical orifice configurations show potential for decreasing the added
drag. However, this depends both on the ellipse dimensions and orientation.
For streamwise-oriented slots, the drag variation strongly depends on the
spanwise size of the orifices. The narrow orifices of flow case $L$-$E_{x-9}$
lead to a substantially lower drag than the baseline case, whereas the wider
slots of case $L$-$E_{x-3}$ result in a massive drag increase. It is
interesting to note that the elliptical slots of $L$-$E_{x-9}$ have a spanwise
width of size $d_{z}^{+}\approx 10$, which is similar to the spacing of drag-
reducing riblets [28]. For spanwise-oriented slots, $\Delta U^{+}$ is less
sensitive to the slots size, and we find the same or a marginally lower drag
than the baseline liner. These findings confirm the experiments of Howerton
and Jones [17] who observed lower drag for spanwise-oriented rectangular
slots. They found that streamwise-oriented rectangular slots increased drag
compared to the baseline case, however, they did not investigate the effect of
slot size, which might have been too large in viscous units to observe the
‘riblets-like’ effect we report in this study. The tapered orifice also
decreases drag compared to the baseline case. In this case, the improved
performance can be traced back to the reduced superficial porosity experienced
by the flow.

Figure 4: Streamwise velocity fluctuations on a wall-parallel plane at $y^{+}+\ell_{T}^{+}=8$ for cases $S$ (a), $L$-$S$ (b), $L$-$T$ (c), $L$-$E_{z-3}$ (d), $L$-$E_{z-6}$ (e), $L$-$E_{z-9}$ (f), $L$-$E_{x-3}$ (g), $L$-$E_{x-6}$ (h), $L$-$E_{x-9}$ (i). Figure 5: Wall-normal velocity fluctuations on a wall-parallel plane at $y^{+}+\ell_{T}^{+}=8$ for cases $S$ (a), $L$-$S$ (b), $L$-$T$ (c), $L$-$E_{z-3}$ (d), $L$-$E_{z-6}$ (e), $L$-$E_{z-9}$ (f), $L$-$E_{x-3}$ (g), $L$-$E_{x-6}$ (h), $L$-$E_{x-9}$ (i). (f) |  (b)
---|---
Figure 6: Wall normal Reynolds stress $\tau_{22}$, as a function of $y^{+}$ (a) and as a function of $y/t$ (b). Line types and symbols are as in Fig. 3. (a) |  (b)
---|---
Figure 7: Wall normal Reynolds stress $\tau_{22}$, as a function of $y^{+}$
(a) and as a function of $y/t$ (b). Line types and symbols are as in Fig. 3.

In a previous study, we related the added drag induced by acoustic liners to
the wall-normal velocity fluctuations [4]; thus, we inspect instantaneous
velocity realizations in wall parallel planes at $y^{+}+\ell_{T}^{+}=8$ in
Fig. 4 and 5. For all liner cases, we note that near-wall streaks are
disrupted and much shorter than on a smooth wall, see Fig. 4. The break-up of
the near-wall streaks is particularly evident for case $L$-$E_{x-3}$ (Fig. 4
(g)), where the near-wall flow deviates substantially from the typical
organization found for the smooth wall (Fig. 4 (a)).

For the baseline liner geometry, wall-normal velocity fluctuations are
concentrated, primarily, around the orifice location, and the positions of the
orifices are clearly visible in the contours of the wall-normal velocity, see
Fig. 5 (b). This effect is more evident for cases $L$-$E_{x-3}$ and
$L$-$E_{x-6}$, where the wall-normal velocity fluctuations in the near wall
region originate primarily from the orifices. The higher wall-normal velocity
fluctuations for flow case $L$-$E_{x-3}$ are correlated with the significantly
higher drag of this case.

Furthermore, we find that liner cases with lower drag present lower wall-
normal velocity fluctuations and a more smooth-wall-like organization of the
near-wall flow. These qualitative observations suggest that the correlation
formulated by Shahzad et al. [4] that relates the added drag and the intensity
of the wall-normal velocity fluctuations holds for all facesheet geometries
under scrutiny.

For a more quantitative analysis, we further show the wall-normal Reynolds
stress components
$\tau_{22}=\overline{\rho}\widetilde{v^{\prime\prime}v^{\prime\prime}}$ in
Fig. 6 and 7. It is clear that the wall-normal velocity fluctuations are not
zero, irrespective of the case considered, above and below the facesheet. We
find that flow cases that exhibit lower drag than the baseline liner show
lower wall-normal velocity fluctuations in the near-wall region and below the
facesheet, Fig. 6. The converse is true for cases that increase the added
drag, Fig. 7.

###  3.2 Acoustic Attenuation

Figure 8: Instantaneous pressure fluctuations over the smooth wall (a) and the baseline liner (b). $p_{0}$ is the reference thermodynamic pressure. $x_{s}$ is the streamwise location where the liners start. The frequency of the sound waves is $f/f_{r}=1$. Figure 9: Instantaneous pressure fluctuations for the tapered orifice configuration for various acoustic wave frequencies $f/f_{r}$= $0.5$ (a), $0.75$ (b), $1.0$ (c) and $1.5$ (d). (a) |  (b)
---|---
Figure 10: Sound pressure level as a function of the streamwise distance from the start of the liners. Line types and symbols are as in Fig. 3. (a) |  (b)
---|---
Figure 11: Sound pressure level as a function of the forcing frequency. Line
types and symbols are as in Fig. 3.

We have studied the aerodynamic drag of different orifice configurations and
have identified several geometries that induce lower drag compared to the
baseline liner with circular orifices. As acoustic liners are used to reduce
engine noise, it is imperative to also test their acoustic performance. In
order to do so, we study the noise attenuation properties of these novel
configurations in the absence of grazing flow. A complete picture of the noise
attenuation properties of aerodynamically optimised acoustic liner geometries
would require including the influence of grazing flow. However, studies of
acoustic liners without grazing flow have been extensively used to provide an
initial estimation of their acoustic performance [33, 34, 35]. Simulations in
the absence of grazing flow allow us to study the acoustic performance over a
range of frequencies for all geometries considered at tractable computational
cost.

Simulations are performed in a channel of size $L_{x}\times L_{y}\times
L_{z}=1250d\times(25d+h)\times 4.69d$, where $d$ is the orifice diameter of
the baseline liner and $h=6.25d$ is the depth of the liner. The spanwise
domain size corresponds to a single cavity, and periodic boundary conditions
are applied in the spanwise direction. We apply no-slip isothermal boundary
conditions on the top and bottom walls and place acoustic liners at the bottom
wall of the channel. An array of $40\times 1$ acoustic liner cavities is
placed between $x_{s}=500d$ and $x_{e}=687.5d$. The geometries of the liners
are the same as those considered when studying the aerodynamic performance.
The resonance frequency of the system is expected to be approximately
$f_{r}\approx 0.028c/d$. However, the correction due to the pressure field,
particularly for the geometries considered in the paper, is difficult to
estimate. We, therefore, test the performance of the novel geometries over a
range of frequencies from $f/f_{r}=0.5$ to $f/f_{r}=1.5$ at a Reynolds number
$Re_{c}=800$ based on the orifice diameter and speed of sound. We verified
that, while the absolute sound attenuation of these liners changes, the
relative sound attenuation, compared to the baseline acoustic liner, does not
depend on the Reynolds number.

Figure 8 shows an instantaneous snapshot of the pressure for the smooth wall
case and the baseline liner, for the case with $f=f_{r}$. Figure 9 also shows
how the instantaneous pressure field changes as the forcing frequency is
changed. As the frequency of the acoustic wave is tuned to the resonant
frequency of the liner, we see a significant attenuation of the acoustic wave
over the liner. The amplitude of the pressure fluctuations towards the end of
the acoustic liner is much lower than over the smooth wall, where the fluid
viscosity is the only dissipative mechanism.

Figure 10 compares the SPL evolution of the acoustic wave for all the liner
configurations and the smooth wall, for the case with $f=f_{r}$. Although
there is a clear SPL reduction for all configurations as compared to the
smooth wall, some geometries perform better than others when evaluated against
the baseline liner. This is true for the SPL loss over the entire frequency
range considered, see Fig. 11. Cases $L$-$E_{x-3}$ and $L$-$E_{z-3}$, offer
very similar acoustic noise attenuation over the entire frequency range,
however, the narrow orifices of cases $L$-$E_{x-6}$, $L$-$E_{z-6}$,
$L$-$E_{x-9}$ and $L$-$E_{z-9}$, that helped reduce drag, appear to inhibit
acoustic noise attenuation. Cases $L$-$E_{x-6}$ and $L$-$E_{z-6}$ show similar
noise attenuation properties at frequencies other than $f=0.75f_{r}$, whereas
cases $L$-$E_{x-9}$ and $L$-$E_{z-9}$ show significantly lower attenuation
throughout the frequency range. The tapered hole configuration provides
comparable acoustic noise attenuation as compared to the standard acoustic
liner. Figure 11 also shows that the effective resonance frequency is about
25% lower than the nominal one, that we estimated disregarding entrance and
exit effects. Notably, the resonant frequency appears to be about the same for
all orifice shapes, suggesting that the required correction for entrance and
exit effects is similar for all orifice shapes. The tapered holes, therefore,
improve not only the aerodynamic performance of the liner but also perform
well acoustically. The elliptical orifices, however, depending upon the width
of the orifice, may or may not have a detrimental influence on acoustic noise
attenuation.

##  4 Conclusion

We have proposed several novel orifice geometries for acoustic liners, some of
which reduce aerodynamic drag without compromising the acoustic performance.
The idea behind the aerodynamically optimized geometries is based on flow
physics, which allows us to restrict the vast parameter space that one could
explore. The aerodynamic performance of the novel geometries is scrutinised
based on Direct Numerical Simulations of fully resolved acoustic liner arrays
in a turbulent channel flow. We find that tapered circular orifices minimize
drag compared to a baseline acoustic liner by reducing the apparent porosity
at the surface of the facesheet, whereas replacing the circular holes with
elliptical slots can lead to a substantially lower drag if the minor axis of
the ellipse is sufficiently thin in viscous units. We find that despite the
very different configurations tested, all optimised geometries work by
reducing the interaction of the flow above and below the surface of the
facesheet, which is confirmed by the reduced wall-normal velocity fluctuations
in the proximity of the facesheet.

We also test the acoustic performance of the proposed liner configurations and
find that the thin elliptical slots have a substantially reduced acoustic
performance compared to a baseline liner; therefore, the improved aerodynamic
benefit comes at the cost of lower noise reduction. The tapered hole
configuration has slightly better noise attenuation properties than the
baseline liner while offering substantially lower aerodynamic drag, and it is
therefore superior both from an aerodynamic and acoustic perspective. In
addition, tapered holes are easy to manufacture and therefore represent a
viable modification to be implemented in existing designs.

## Acknowledgments

We acknowledge EuroHPC for awarding us access to Meluxina, at LuxProvide,
Luxembourg.

Data availability statement

DNS data is available at
https://doi.org/10.4121/1b59fc2b-d837-4479-836e-810aa5b0db3d

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