Article
Sound propagation in a
lined nozzle with shear
and bias flows
International Journal of Aeroacoustics
0(0) 1–46
! The Author(s) 2018
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DOI: 10.1177/1475472X18812795
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LMBC Campos1 and C Legendre2
Abstract
In this study, the propagation of waves in a two-dimensional parallel-sided nozzle is considered
allowing for the combination of: (a) distinct impedances of the upper and lower walls; (b) upper
and lower boundary layers with different thicknesses with linear shear velocity profiles matched
to a uniform core flow; and (c) a uniform cross-flow as a bias flow out of one and into the other
porous acoustic liner. The model involves an “acoustic triple deck” consisting of third-order nonsinusoidal non-plane acoustic-shear waves in the upper and lower boundary layers coupled to
convected plane sinusoidal acoustic waves in the uniform core flow. The acoustic modes are
determined from a dispersion relation corresponding to the vanishing of an 8 8 matrix determinant, and the waveforms are combinations of two acoustic and two sets of three acousticshear waves. The eigenvalues are calculated and the waveforms are plotted for a wide range of
values of the four parameters of the problem, namely: (i/ii) the core and bias flow Mach numbers;
(iii) the impedances at the two walls; and (iv) the thicknesses of the two boundary layers relative
to each other and the core flow. It is shown that all three main physical phenomena considered in
this model can have a significant effect on the wave field: (c) a bias or cross-flow even with small
Mach number M0 0:03 0:06 relative to the mean flow Mach number M1 0:3 1:2 can
modify the waveforms; (b) the possibly dissimilar impedances of the lined walls can absorb (or
amplify) waves more or less depending on the reactance and inductance; (a) the exchange of the
wave energy with the shear flow is also important, since for the same stream velocity, a thin
boundary layer has higher vorticity, and lower vorticity corresponds to a thicker boundary layer.
The combination of all these three effects (a–c) leads to a large set of different waveforms in the
duct that are plotted for a wide range of the parameters (i–iv) of the problem.
1
Center for Aeronautical and Space Science and Technology (CCTAE), Instituto Superior Técnico (IST), Technical
University of Lisbon, Lisboa Codex, Portugal
2
Free Field Technologies S.A, Mont-Saint-Guibert, Belgium
Corresponding author:
C Legendre, Free Field Technologies S.A., Mont-Saint-Guibert, Belgium.
Email: Cesar.Legendre@fft.be
2
International Journal of Aeroacoustics 0(0)
Keywords
Shear flow, cross flow, liners, duct modes
Date received: 10 October 2017; accepted: 22 June 2018
Introduction
The propagation of sound in the inlets and nozzles of jet engines is one of the major
problems in aeroacoustics.1–7 The shear layers near the walls lead to the coupling of
sound with vorticity significantly changing the properties of the acoustics waves.8–25 In
order to reduce noise, the nozzle walls often use liners that, if locally reacting, can be
represented by an impedance distribution.26–41 The bias flow out perforated liners can
have a significant effect on sound in a boundary layer42 even for small velocities. The
boundary layer may be modeled as a “double deck,” with a shear flow matched to a uniform
stream. The present paper extends the model to a nozzle represented by a “triple deck”
consisting of two possible dissimilar wall shear flows matched to a uniform core flow, with
bias cross-flow and locally reacting walls. Thus, the present model embodies (Figure 1) the
following features: (i) a plane parallel-sided nozzle with a uniform core flow, matched by
two possible dissimilar boundary layers to zero velocity at the walls; (ii) the two boundary
layers have possibly different linear velocity profiles matched to the free stream; (iii) both
walls are locally reacting with possible distinct impedances; (iv) there is a uniform bias flow
orthogonal to the shear flow.
The main difference between the present paper and the earlier work42 is that: (i) the
earlier work42 considered acoustic-vortical waves propagating in a boundary layer with
Figure 1. Shear and cross flow in a parallel-sided 2D nozzle.
Campos and Legendre
3
a linear velocity profile, matched to a uniform free stream, in the presence of a uniform bias
or cross-flow, leading to a “double deck”; (ii) the present paper considers a “triple deck,”
that is standing acoustic-vortical waves in a parallel-sided duct with distinct wall impedances, and two possibly dissimilar boundary layers, matched to the same free stream, also in
the presence of a bias flow. The present paper also addresses two issues raised in the literature43 about the early work.42 The short derivation of the mean flow42 has been misinterpreted43 leading to the claim that it is inconsistent, unless an “artificial heat input” is
imposed; the present paper starts with a more detailed derivation of the homentropic
mean flow, showing that it is consistent with no need for heat input. The second issue is
the claim43 that the wave equation42 is incorrect, offering as substitute a relation among four
wave variables that cannot be compared to a proper wave equation in one variable. The
detailed derivation of the consistent homentropic mean flow in the first section of the present
paper is followed in the second section by a detailed proof that the acoustic-vortical wave
equation derived in the earlier work42 is correct. In conclusion, the comparison of
the present and earlier42 work shows that: (i) the homentropic mean shear flow and
the acoustic-vortical wave equation are the same, with a more detailed derivation
to avoid possible ambiguities; (ii) the same acoustic-vortical wave equation is applied to
distinct problems, namely “double deck” with free waves in a semi-infinite medium42 versus
a “tripple-deck” with duct modes in a parallel-sided nozzle of finite width in the present paper.
Thus, the present paper considers a parallel-sided nozzle with both boundary layers of a
linear shear flow to a free stream and with superimposed uniform cross-flow leading to a
“triple deck.” The combined linear shear and bias flow in each boundary layer of the “triple
deck” are similar to the “double deck” in Campos et al.42 It has been argued43 that this mean
flow requires heat addition; therefore, a detailed derivation of this mean flow is given in the
next sections, providing that the mean flow is consistent in homentropic conditions. The
main effect of the bias flow is to raise the order of the acoustic-vortical wave equation from
two to three, eliminating also the critical layer for zero Doppler-shifted frequency42; since
this wave equation was questioned,43 a detailed derivation is also provided. Afterwards the
sound fields in the two boundary layers are matched to the core flow leading to a 8 8
system of equations consisting of: (i) three matching conditions at the edge of each layer
since the wave equation is of the third-order; (ii) one impedance boundary condition at each
wall, for a total of eight conditions. The eigenvalues of an 8 8 determinant describe the
wave modes and the eigenfunctions specify the waveforms. The amplitude and phase of
the wave pressure perturbation are plotted (Figures 4 to 27) for several combination of the
dimensionless parameters of the problem, namely: (i–ii) the core flow and bias flow Mach
numbers; (iii) the Helmholtz number of dimensionless frequency based on the constant
sound speed at the wall distance; (iv–v) the possibly dissimilar thicknesses of the two
shear layers as a fraction of the nozzle width; (vi–vii) the possibly distinct impedance at
the two walls.
The main overall purpose of the paper is to investigate three effects on the acousticvortical wave field in the parallel-sided nozzle: (a) the possibly dissimilar impedances on the
two walls with locally reacting liners, that can absorb (or amplify) waves more or less; (c) the
bias flow that can considerably modify the waveforms, ever for small Mach numbers relative
to the Mach number of the free stream; (b) the exchange of wave energy with the shear flow
in the possibly dissimilar boundary layers.
4
International Journal of Aeroacoustics 0(0)
A consistent combination of shear and cross flows
In Campos et al.,42 a mean flow velocity is assumed in the following form
8
<U
y
ex þ V0 ey
v0 ðyÞ ¼
L
: U1 ex þ V0 ey
1
for 0 y L ð1aÞ
for L y < 1
ð1bÞ
that is continuous at the junction between the linear boundary layer (1a) and uniform free
stream (1b), and consists of: (i) a uniform cross-flow V0; (ii) a linear shear layer parallel to
the wall U1 y=L matched to a uniform flow U1 in the free stream. The following analysis
starts from a more general free stream velocity
v0 ðx; yÞ ¼
UðyÞ ex þ VðxÞ ey
U1 ex þ V0 ey
for 0 y L
for L y < 1
ð2aÞ
ð2bÞ
and shows that for a homentropic flow under a certain approximation
0 y L : UðyÞ ¼ U1
y
;
L
VðxÞ ¼ V0
(3a-c)
the mean flow velocity (2a,b) simplifies to (1a,b).
The mean flow in the boundary layer (2a) consists of two shears one in the x and the
other in the ydirection. They must satisfy the steady non-uniform mean flow equations of
continuity (4a), momentum (4b) and adiabaticity (4c)
r ðq0 v0 Þ ¼ 0;
rp0 þ q0 ðv0 rÞv0 ¼ 0;
v0 rp0 ¼ c20 v0 rq0
(4a-c)
where p0 is the pressure, q0 the mass density and c0 is the adiabatic speed of sound. The
adiabatic equation (4c) is satisfied by equation (5b) with constant speed of sound c0 and
adiabatic exponent c
c; c0 ¼ const: : cp0 ðx; yÞ ¼ c20 q0 ðx; yÞ
(5a,b)
Furthermore, the equation of continuity (4a) using the mean flow the boundary layer (2a)
leads to (6a)
@q
@q
dy
@q =@x VðxÞ
¼
UðyÞ 0 þ VðxÞ 0 ¼ 0 :
(6a,b)
¼ 0
@x
@y
dx q0
@q0 =@y UðyÞ
which shows that the mass density is conserved along the isocores that coincide with the
streamlines (6b). Besides, the equation of momentum (4b) has, respectively, the following x; ycomponents
@p0
dU
¼ q0 ðx; yÞVðxÞ
;
@x
dy
@p0
dV
¼ q0 ðx; yÞUðyÞ
@y
dx
(7a,b)
Campos and Legendre
5
from which it follows that the pressure is constant along the isobars (8a)
dy
dx
p0
@p0 =@x
VðxÞdU=dy
¼
;
¼
@p0 =@y
UðyÞdV=dx
dy
dx
¼
p0
dy
dx
(8a,b)
q0
Also equation (5a,b) implies that the mass density and pressure vary along the same lines
(8b), that is isobars, isocores and streamlines all coincide.
Therefore, the equations of continuity, momentum and adiabaticity (4a-c) are satisfied by
the velocity field in the boundary layer (2a) provided that the isocores coincide with the
isobars (8b) implying from equations (6b) to (8a) that the mean flow velocity components U,
V have the same shear rate - but with opposite signs (9a,b)
dU
¼ -;
dy
dV
¼ -;
dx
UðyÞ ¼ U0 þ -y;
VðxÞ ¼ V0 -x
(9a-d)
and since the l.h.s. of (9a) depends only on y and the l.h.s. of (9b) only on x, the shear must be constant, leading to the horizontal U and vertical U components of the mean flow
(9c,d). Assuming a zero tangential velocity at the wall (10a), the horizontal velocity (10b)
coincides with a linear shear boundary layer (3b) specifying the shear rate (10c)
Uð0Þ ¼ U0 ¼ 0 : UðyÞ ¼ -y ¼ U1
y
;
L
-
U1
L
(10a-c)
Substitution of the horizontal and vertical mean flow components U, V (9c,d) with the
consideration of a linear shear boundary layer for the horizontal component (10b) in the
x; ycomponents of the momentum, equations (7a) and (7b) specify the components of
the pressure gradient
c20 @p0
¼ -ðV0 -xÞ;
cp0 @x
c20 @p0
¼ -2 y
cp0 @y
(11a,b)
The previous relation follows the exact differential for the mean flow pressure (12a)
whose integral is (12b)
c20
dp0 ¼ -V0 dx þ -2 ðxdx þ ydyÞ
cp0
"
#
p0 ðx; yÞ
c-V0
c-2 2
¼ exp
x þ 2 x þ y2
p0 ð0; 0Þ
c20
2c0
(12a)
(12b)
Subsequently, since the sound speed and adiabatic exponent are constant (5a), the pressure p0 and mass density q0 have the same spatial dependence and using (10c) this leads to
"
#
q0 ðx; yÞ p0 ðx; yÞ
cU1
x2 þ y2
¼
¼ exp 2
V0 x U 1
2L
q0 ð0; 0Þ p0 ð0; 0Þ
c0 L
(13)
6
International Journal of Aeroacoustics 0(0)
Assumption (14a) suppresses the quadratic term in x2 in equation (13) leading to simplifications in the relations for the mean pressure and density
x
V0 q0 ðx; yÞ p0 ðx; yÞ
cU1 V0 x cU21 y2
¼
¼ exp þ 2 2
:
U1 q0 ð0; 0Þ p0 ð0; 0Þ
L
c20 L
2c0 L
!
(14a,b)
The assumption (14a) limits the distance along the wall to a fraction of the thickness of the
boundary layer smaller that the ratio of the bias flow velocity to the free stream velocity. The
assumption (14a) is not needed to satisfy the energy balance, because it is already implicitly
satisfied by the fundamental equations (4a–c) from which equation (13) was derived; the
assumption (14a) is a first condition for the simplification of the mean flow pressure gradient, that is considered further in the sequel. Bearing in mind the limited distance from the
wall in the boundary layer (3a), the last term in (14b) is negligible for low Mach number
(15b) of the free stream (15a) leading to (15c)
M1 ¼
U1 2
q ðx; yÞ p0 ðx; yÞ
cM1 V0 x
¼
¼ exp M1 1 : 0
c0
c0 L
q0 ð0; 0Þ p0 ð0; 0Þ
(15a,b)
Usually the bias or cross flow velocity V0 does not exceed the free stream velocity U1
implying the same (16b) for the Mach numbers, i.e. free stream (15a) and cross flow (16a),
and then the limited distance from the wall (14a) implies (16c). In addition, the low free
stream Mach number (15b) together with (16b) implies (16d). The inequalities (16c,d) show
that the argument in the exponential in (15c) is small leading to a ratio of the mean flow
pressure (or density) to its value at the origin close to unity (16e)
M0 ¼
V0
M1
c0
M0 ;
x
1;
L
M0 M1 1 :
q0 ðx; yÞ p0 ðx; yÞ
¼
1
q0 ð0; 0Þ p0 ð0; 0Þ
(16a-d)
This proves that the mean flow pressure and the mass density are constant under the
assumption of: (i) low (15b) Mach number (15a) of the free stream; (ii) cross flow velocity or
Mach number (16a) not exceeding (16b) that of the free stream; (iii) small distance along the
wall compared with the boundary layer thickness (16c) or (14a). The mean flow pressure
(15c) can be written to lowest order in the argument
cU1 V0 x
p0 ðx; yÞp0 ð0; 0Þ 1 c20 L
(17)
Consequently, the condition of constant mean flow pressure (18a) requires (18b)
p0 ðx; yÞ ’ const: :
x
c20
1
x
¼
cU1 V0 cM1 M0
L
1
(18a-c)
Campos and Legendre
7
which amply met equation (18c) on account of (16d). Condition (18b) can be written in the
equivalent and alternative form (18d) to (18f)
x
c20 L
p0 L
p0
¼
¼
cU1 V0 q0 U1 V0 q0 V0 -
(18d-f)
that coincides with equation (2.12) in Campos et al.42 The condition of constant mean flow
pressure, density and sound speed simplifies the derivation of the acoustic-vortical wave
equation in a linear shear flow in the presence of a cross-flow in the next section. This is the
reason to focus on the mean flow pressure gradient being small.
Similarly, the pressure gradient normal to the wall inside the boundary layer can be
derived from equation (13) and is given by
0<y L:
@p0
cU2 y
¼ p0 ðx; yÞ 2 12 ;
@y
c0 L
@p0
p0 ðx; yÞ 2
c
M1
@y
L
(19a-c)
showing that in the boundary layer, the transversal pressure gradient (19b) has the upper
bound (19c) involving the small (15b) free stream Mach number (15a). Also, the pressure
gradient along the wall can be derived from equation (13) that simplifies to equation (20b)
using the fact that the distance along the wall is limited (14a)
@p0 ðx; yÞ
cU1
x
cU1 V0
p0 ðx; yÞ
¼ p0 ðx; yÞ 2
p0 ðx; yÞ 2
M0 M1
V 0 U1
¼c
@x
L
L
c0 L
c0 L
(20a-c)
where it can be observed that the longitudinal pressure gradient (20c) involves the small
product of the free stream and cross flow Mach numbers (16d). And the substitution of the
constant shear rate (10c) in the vertical mean velocity V (9d) leads to the constant cross-flow
velocity (21b) under the approximation (21a) that coincides with (14a), i.e. limited distance
along the wall
x
V0
U1
U1 x
x ¼ V0 1 V0
: VðxÞ ¼ V0 (21a,b)
U1
L
V0 L
L
Thus, the mean flow (2a,b) with the approximation of a limited distance along the wall,
i.e. (14a)(21a), leads to: (i) the mean flow (1a,b) that coincides with equations (2.6a,b) in
Campos et al.42; (ii) exactly constant sound speed and adiabatic exponent (5a); (iii) pressure
and density gradients (20c,19b) can be rewritten as follows
rp0 rq0 cM1
y
M1 ey M0 ex
¼
p0
q0
L
L
(22)
The previous equation is the result of the approximation implying a limited distance
along wall distance (14a) applied to (20a,19b)
rp0 rq0 cM1
y
x
M1 ey þ M1 M0 ex
¼
¼
p0
q0
L
L
L
(23)
8
International Journal of Aeroacoustics 0(0)
The gradients of the mean pressure and mass density are small (22), leading to approximately constant values (16e) if in addition to the assumption of limited distance along the
wall (14a) are assumed low (15b,16b) free stream (15a) and cross-flow (16a) Mach numbers
and also their cross product (16d).
The consistency of the boundary layer and the free stream mean flow represented by
equations (1a,b) can be confirmed by writing the velocity components
u0 ðx; yÞ ¼
U1
½L þ ðy LÞHðL yÞ ;
L
v0 ðx; yÞ ¼ V0
(24a,b)
where HðnÞ is the Heaviside unit function of argument n, substituting equation (24a,b) in the
momentum equation (4b)
u0
@u0
@u0
1 @p0
þ v0
¼
;
@x
@y
q0 @x
u0
@v0
@v0
1 @p0
þ v0
¼
@x
@y
q0 @y
(25a,b)
and using the adiabatic equation with constant speed of sound and adiabatic exponent (5a,b)
leads to the following components of the pressure gradient (26b,c)
1 cV0 U1 @p0
p0
¼ HðL yÞ;
¼
:
2
@x
l
l
c0 L
@p0
¼ 0;
@y
(26a-c)
involving the constant l specified by equation (26a). The derivation of the xcomponent of
the pressure gradient (26b) follows from the combination of the horizontal velocity (24a)
with the x ¼component of the momentum equation (25a)
@p0
U1
¼ q0 V0
½ðy LÞdðL yÞ þ HðL yÞ ;
@x
L
@p0
p 0 V 0 U1 c
¼
HðL yÞ
@x
c20 L
(27a,b)
and uses the following properties of the Heaviside unit function H and the Dirac delta function
.¼yL:
dH
¼ dð.Þ;
d.
. dð.Þ ¼ 0
(28a-c)
In the free stream, where the velocity is constant, the pressure gradient (26b,c) is zero
(29b-c) leading to a constant pressure
y>L:
@p0
@p0
¼0¼
) p0 ðx; yÞ ¼ const
@x
@y
(29a-d)
On the other hand, in the boundary layer, the pressure gradient (26b,c) simplifies to (30b,
c) implying (30d)
@p0
p0
¼ ;
0<y L:
@x
l
@p0
p0 ðx; yÞ
x
cV0 U1 x
¼0)
¼ exp ¼ exp @y
p0 ð0; 0Þ
l
c20 L
(30a-d)
Campos and Legendre
9
The approximation (18b), that coincides with equation (2.12) in Campos et al.,42
when applied to (30d) leads to a constant pressure also in the boundary layer, i.e. (18a)
(29d). Both proofs lead to an exactly adiabatic mean flow that does not require any
heat input. This mean flow is used next to confirm the wave equation (2.25) in
Campos et al.42
Third-order wave equation for the pressure perturbation
The total flow, consisting of the mean flow plus acoustic perturbation satisfies the continuity
(31a), inviscid momentum (31b) and adiabatic (31c) equations
D~
q
~ ðr ~v Þ ¼ 0;
þq
Dt
~
q
D~v
þ r~
p ¼ 0;
Dt
D~
p
D~
q
¼ c2
Dt
Dt
(31a-c)
where D/Dt is the exact material derivative (32a)
D
@
þ ~v r;
Dt @t
d
@
þ v0 r
dt @t
(32a,b)
and it is distinct from the linearised material derivative (32b). The perturbations around the
mean flow are written as
v~ ¼ V0 þ vðx; y; tÞ;
u~ ¼ U0 ðyÞ þ uðx; y; tÞ;
p~ ¼ p0 þ pðx; y; tÞ;
~ ¼ q0 þ qðx; y; tÞ
q
(33a-d)
and replaced in the fluid dynamics where equations (31a-c) lead to the linearised equations
of continuity (34a), momentum (34b) and adiabaticity (34c) to
dq
þ q0 r v þ v rq0 ¼ 0
dt
dv
q0
þ ðv rÞv0 þ qðv0 rÞv0 þ rp ¼ 0
dt
(34a)
dp
2 dq
þ v rp0 ¼ c0
þ v rq0
dt
dt
(34b)
(34c)
The previous equations (34a-c) are eliminated next to obtain a scalar wave equation for
the pressure perturbation. The linearised material derivative (32b) is given in the boundary
layer with cross flow (1a) by
d
@
@
@
@
þ v0 :r ¼ þ U0 ðyÞ
þ V0 :
dt @t
@t
@x
@y
(35)
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International Journal of Aeroacoustics 0(0)
Moreover, the three terms of the linearised continuity equation (34a) scale as
dq
xq v
xq 0 ;
dt
c0
q0 r vq0 kv
q0 xv
;
c0
v rq0 q0 vM1
fM1 ; M0 g
L
(36a-d)
where: (i) in equation (36a), the frequency d=dtx and q=q0 v=c0 were used; (ii) in equation
(36b), the wavenumber rkx=c0 was used; (iii) in equation (36c,d), equation (22) and vu
were used. The first two terms in equation (34a) are comparable to equations (36a,b)
between each other, and the last can be neglected (37c) if the conditions (37b,c) are met
M21 X
M0 M1 :
M21 ; M0 M1
v rq0 v rq0 c0
M1 fM1 ; M0 g
1
q0 r v dq=dt xL
X
(37a-c)
The restrictions (37a,b) imply that the wave equation to be derived using this flow model
applies to higher Helmholtz and smaller Mach numbers. In the particular case of low free
stream and cross-flow Mach numbers (15a,16c), i.e. M21 1 and M1 M0 1, the conditions (37a,b) are met for Helmholtz numbers of order unity or larger X 1. The ray
approximation of the wavelength k0 much larger than the thickness of the boundary layer
1, whereas the present theory is more
L corresponds to a large Helmholtz number X2
general including wavelength comparable to the boundary layer thickness, when the interaction between the sound and the shear flow should be stronger
X
xL
2pL
k0 L ¼
ⲏ1
c0
k0
(38a-c)
The compactness approximation X 1 is excluded by condition (38c), and in this case of
wavelength large compared with the thickness of the boundary layer, the latter should have
a small effect on sound.
To continue to the derivation of the wave equation, the approximations (37c) simplify the
linearised equation of continuity from equations (34a) to (39a) and for an adiabatic mean
flow, equation (5a,b) holds equation (39b)
dq
þ q0 r v ¼ 0;
dt
v rp0 ¼ c20 v rq0
(39a,b)
Substitution of (39b) simplifies the adiabatic equations (34c) to (40a) where (39a) may be
used leading to (40b)
dp
dp
@v
2 dq
2 @u
c0
¼ 0;
þ q0 c 0
þ
¼0
(40a,b)
dt
dt
dt
@x @y
Furthermore, the linearised material derivative (35) commutes (41a) with the
xcomponent gradient but not with the y-component gradient (41b)
@ d
d @
¼
;
@x dt dt @x
@ d
d @
dU0 @
¼
þ
dy @x
@y dt dt @y
(41a,b)
Campos and Legendre
11
From equation (4a), it follows
dq0
¼ v rq0 ¼ q0 r v0 ¼ 0
dt
(42)
and using also a constant speed of sound c0 (5a) when applying d/dt to equation
(40b) gives
1 d2 p
@ du
@ dv
dU0 @v
þ
q
¼ q0
þ
q
0
0
dy @x
c0 2 dt2
@x dt
@y dt
(43)
where the commutation relations (41a,b) were used. On the other hand, the x and
ycomponents of the linearised momentum equation (34b) are
q0
du
dU0
dU0 @p
þ qV0
þ
þ q0 v
¼ 0;
dy
dy
dt
@x
q0
dv @p
þ
¼0
dt @y
(44a,b)
Substituting equation (44a,b) in equation (43) gives
1 d2 p
1
dU0 @v
dU0
@ q
q
þ
V
q
r
rp
¼
2q
0
0
0
dy @x
dy 0 @x q0
c0 2 dt2
q0
(45)
The l.h.s of equation (45) is the convected wave equation for pressure, which is
valid when the r.h.s is zero, i.e. in the absence of shear flow. In the presence of shear
flow in order to eliminate v and q from the r.h.s of equation (45) and to obtain a wave
equation for the pressure alone, the linearised material derivative d/dt is applied once more
leading to
1 d3 p
d
1
dU0 @ dv
dU0
@ 1 dq
q
r
þ
V
q
rp
¼
2q
0
0
0
dy @x dt
dy 0 @x q0 dt
c0 2 dt3
dt
q0
(46)
where equation (47b) was used on account of (47a), i.e. a linear shear boundary layer
y d dU0
d2 U0
¼ V0
¼0
UðyÞ ¼ U1 :
dy2
L dt dy
(47a,b)
Substitution of (44b) and (40a), respectively, on the first and second terms on the r.h.s. of
equation (46) gives a wave equation for the pressure perturbation
1 d3 p
d
1
dU0
@ 1 @p
V0 dU0
@ 1 dp
q
q
r
¼0
q
rp
þ
2
0
dy 0 @x q0 @y
c0 2 dt3
dt
q0
c20 dy 0 @x q0 dt
(48)
12
International Journal of Aeroacoustics 0(0)
Thus, equation (48) is the acoustic-vortical wave equation for the pressure perturbation
in the presence of cross and shear flow. In the absence of cross-flow (49a), it reduces to the
Pridmore-Brown equation10 most often written for constant mean flow mass density
as follows
!
d 1 d2 p
dU0 @ 2 p
2
¼0
r p þ2
V0 ¼ 0; q0 ¼ const :
2
2
dy @x@y
dt c0 dt
(49a-c)
The usual form of the Pridmore-Brown equation expressed as relation (49c) can be generalized to
"
#
d 1 d2 p
dU0 @ 2 p
2
¼0
r p þ rp:rðlogq0 Þ þ 2
V0 ¼ 0;
2
2
dy @x@y
dt c0 dt
(50a,b)
to account for a mean flow with non-uniform mass density.
The wave equation (48) can be simplified further using again the approximations (37a,b)
as shown next in the presence of cross-flow. Returning to equation (45), it can be
rewritten as
1 d2 p
1
dU0 @v
dU0 @q
dU0 q @q0
þ V0
V0
r2 p þ rq0 rp ¼ 2q0
dy @x
dy @x
dy q0 @x
c0 2 dt2
q0
(51)
The ratio of the third to the first term on the l.h.s. of equation (51) can be evaluated using
(20c) as
1
q0
rq0 rp
1 d2 p
c0 2 dt2
kp cUc12 LV0
0
x2 p
c20
kcU1 V0 cU1 V0 cM1 M0 cM1 M0
¼
1
¼
x2 L
c0 xL
xL=c0
X
(52)
and is negligible by (37b). The ratio between the third and the first term on the r.h.s. of
relation (51) can be evaluated as
@q0
@x
@v
2q0 @x
V0 qq
0
V0 cv0 q0 cVc02UL1
0
2q0 kv
cV20 U1 cV20 U1 cM20 M1 cM0 M0 M1
cM0
2
¼
3
2
X
2
2c0 xL 2xL=c0
2c0 kL
(53)
and is small (37a,b) for small free stream and cross flow Mach numbers (15a,16c). With
these approximations, equation (51) simplifies to
1 d2 p
dU0 @v
dU0 @q
þ V0
r2 p ¼ 2q0
dy @x
dy @x
c0 2 dt2
(54)
Campos and Legendre
13
that coincides with equation (2.22) in Campos et al.42 Applying d/dt and using the linearised
adiabatic equation (40a) and the y component of the linearised momentum equation (44b)
leads to
"
#
d 1 d2 p
dU0
@2p
V0 @ dp
2
2q0
r p þ
¼0
dy
dt c0 2 dt2
@x@y c20 @x dt
(55)
that coincides with equation (2.25) in Campos et al.42
Sound in a duct with shear and cross flow
The acoustic field in a plane parallel-sided duct with a uniform core flow between boundary
layers with linear shear velocity profiles is specified in the boundary layers by a third-order
wave equation that simplifies to the convected wave equation in the core flow.
Triple-deck for a sound in a plane parallel-sided duct nozzle
Consider (Figure 1) a plane parallel duct of width L with locally reacting walls, with possibly
different uniform impedances Zþ (Z ) at the top (bottom) wall. The boundary layers at each
wall are assumed to have a linear profile for shear velocity, and match to a uniform stream
between the boundary layers. Thus, the longitudinal mean velocity parallel to the walls is
8
y
>
>
U1
>
>
<
L
U0 ðyÞ ¼ U1
>
Ly
>
>
>
: U1 L
þ
if 0 y L
ð56aÞ
if L y L Lþ ð56bÞ
if L Lþ y L ð56cÞ
where L (Lþ ) is the thickness of the lower (upper) boundary layer satisfying L þ Lþ L
and U1 is the uniform velocity of the core flow. Superimposed on the shear flow profile
(56a) to (56c), there is a uniform cross-flow in the vertical direction perpendicular to the
walls with velocity V0, leading to the total mean velocity
v0 ¼ V0 ey þ U0 ðyÞ ex
(57)
In the most practical configurations such as inlets and exhausts of jet engines, the bias
flow comes out of the holes in the acoustic liner, hence upward in the lower wall and
downward in the upper wall. The condition of uniform cross-flow in (57) implies that for
V0 > 0, there is a positive bias flow out of the lower liner wall and a negative bias flow into
the upper lined wall (vice-versa for V0 < 0). This may be an advantage of the model in the
sense that it allows the comparison of the effect of positive and negative bias flow at each
wall. Adding the case without bias flow covers all three possibilities: (i) no bias flow; (ii)
positive bias flow leaving the liner; and (iii) negative bias flow entering the liner. In the
presence of cross-flow, the problem is not symmetric relative to the axis of the duct, even for
equal impedances at the walls; on the other hand, in the absence of cross-flow, the problem
is symmetric for equal wall impedances. This should provide a good basis for comparison
14
International Journal of Aeroacoustics 0(0)
and assessment of bias flow and wall impedance effects. The mean flow (56a-c;57) corresponds to a “triple-deck” for the velocity, and the corresponding triple deck for the acoustic
pressure is considered next.
Acoustic wave equation with shear and cross flows
The mean flow depends neither on the time nor on the longitudinal coordinate x along the
lower wall, and thus a Fourier decomposition may be considered
Z
pðx; y; tÞ ¼
Z
þ1
þ1
dx
1
dk Pðy; k; xÞeiðkxxtÞ
(58)
1
where P is the pressure perturbation spectrum for a wave of frequency x and horizontal
wavenumber k is at the vertical position at a distance y perpendicular to the lower wall. The
acoustic-vortical equation in a linear shear flow with a uniform cross-flow is equation (55),
and d/dt the linearised material derivative is equation (35), and substitution of equation (58)
in equation (35) leads to equation (59a)
d
d
¼ x ðyÞ þ V0
;
dt
dy
x ðyÞ ¼ x kU0 ðyÞ
(59a,b)
where equation (59b) is the wave Doppler frequency shifted by the horizontal shear flow
alone. Substituting equation (58) into equation (55) and using equation (59a) follows a cubic
dispersion relation in k and x, and a third-order differential equation specifying the dependence of the wave spectrum on the distance from the wall42,44
2 00
2
2
1 3M
M0 M0 2 1 L3 P000 þ ix
0 L P þ 2ieM1 1 þ M0 3M0 x
2 e2 2ieM0 M1 P ¼ 0
x
þ ix
2
þ M0 e2 LP0
(60)
where prime denotes the derivative with regard to y and the coefficients involve powers of k
and x up to three. Equation (60) involves four dimensionless parameters: (i) the Mach
number of the core flow (61a); (ii) the Mach number of the cross-flow (61b); (iii) the dimensionless frequency or Helmholtz number (61c); and (iv) the dimensionless horizontal wavenumber or compactness (61d)
M1 U1
V0
xL
; M0 ; X
;
c0
c0
c0
e kL
(61a-d)
These specify the Doppler-shifted frequency (59b) in dimensionless form
¼
x
x L L
xL kU1 y
y
¼ ½x kU0 ðyÞ ¼
¼ X eM1
c0
c0
c0
c0
L
(62)
Campos and Legendre
15
that also appears in equation (60). In the uniform core flow, equation (55) reduces to the
convected wave equation with uniform horizontal (61a) and vertical (61b) Mach numbers
for the mean flow.
Acoustic pressure in the boundary layers and core flow
The third-order differential equation (60) for the acoustic-vortical wave pressure spectrum
has no singularities and thus its solution in the lower boundary layer can be expressed as a
Maclaurin series of y/L
Q
X
n
1
y
y
¼
an
L
L
n¼0
(63)
Substitution of equation (63) in equation (60) leads to the recurrence formula for the
coefficients an.42,44 The recurrence formula shows that the coefficients (a0, a1, a2) may be
chosen arbitrarily. Choosing (a0, a1, a2) ¼ (1, 0, 0), (0, 1, 0) and (0, 0, 1) leads to three linearly
independent solutions denoted respectively by Q1 ðy=LÞOð1Þ; Q2 ðy=LÞOðy=LÞ and
Q3 ðy=LÞOðy2 =L2 Þ. The general integral for the wave pressure in the lower boundary
layer (64a) is given by a linear combination (64b) of these functions with arbitrary coefficients (A1, A2, A3)
0 y L :
P ðyÞ ¼ A1 Q1
y
L
þ A 2 Q2
y
L
þ A 3 Q3
y
L
(64a,b)
In the upper boundary layer (65a), the wave pressure is given by a similar expression (65b)
L Lþ y L :
Pþ ðyÞ ¼ B1 Q1
Ly
Ly
Ly
þ B 2 Q2
þ B3 Q3
Lþ
Lþ
Lþ
(65a,b)
involving the same functions of ðL yÞ=ðLþ Þ in (56c) instead of y=L in (56a), with
distinct arbitrary constants (Figure 2). In the core flow, the wave equation (55) reduces to
the convected wave equation with uniform horizontal U1 and vertical V0 mean
flow velocities
(
)
@
@
@ 2
þ U1
þ V0
c0 2 r2 p1 ðx; y; tÞ ¼ 0
@t
@x
@y
(66)
and the acoustic pressure spectrum (58) is a superposition of an upward propagating wave
with amplitude Cþ and vertical wavenumber jþ and a downward propagating wave with
amplitude C and vertical wavenumber j
L y L Lþ :
P1 ðy; k; xÞ ¼ Cþ eijþ y þ C eij y
(67a,b)
16
International Journal of Aeroacoustics 0(0)
Figure 2. Acoustic lined duct in presence of shear and cross flow.
where j are the vertical (68b) and k is the horizontal (68a) wavenumbers, respectively
k¼
x
cos h; j ¼
c0
M1 k x=c0 sgnðM0 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx=c0 M1 kÞ2 k2 ð1 M20 Þ
1 M20
(68a,b)
in terms of the angle h of propagation with the horizontal axis and the sign of the cross flow
Mach number sgnðM0 Þ. By continuity, the horizontal wavenumber (68a) has the same value
in the core flow and the boundary layers; it can be used to calculate the corresponding
dimensionless form or compactness (61d). The vertical wavenumber (68b) exists only for
acoustic waves in the uniform core flow (67a,b) and does not apply to acoustic-vortical
waves in the boundary layers (64a,b;65a,b) where the acoustic pressure (63) is not a sinusoidal function of the distance from the wall.
Determination of the duct modes
The impedance boundary conditions at the duct walls together with the matching of the
wave pressure at the junction of the core flow with the boundary layers specify the eigenvalues for the horizontal wavenumber as the roots of an 8 8 matrix determinant.
Impedance boundary condition for locally reacting liners
Both the lower and upper walls are assumed to be covered by locally reacting liners leading
to an impedance boundary condition, e.g. for the lower wall
P ð0; k; xÞ ¼ Z ðxÞV ð0; k; xÞ
(69)
Campos and Legendre
17
relating the pressure (58) and the vertical velocity (70b) wave perturbation spectra
Z
u; vðx; y; tÞ ¼
Z
þ1
þ1
dx
1
dk U; Vðy; k; xÞeiðkxxtÞ
(70a,b)
1
The ycomponent of the linearised momentum equation (44b) leads to (71) for the
wave spectra
1 0
P ¼ ixV þ ikU0 ðyÞV þ V0 V0
q0 (71)
At the lower wall (72a), this leads to the impedance boundary condition (72b)
U0 ð0Þ ¼ 0 : P0 ð0; k; xÞ ¼ q0 ixV ð0; k; xÞ V0 V0 ð0; k; xÞ
(72a,b)
Substituting V from equation (69) in equation (72b) leads to the impedance boundary
condition at the lower wall in terms of the wave pressure spectrum alone (73)
q V0
ixq0
¼
P ð0; k; xÞ
P0 ð0; k; xÞ 1 0
Z ðxÞ
Z ðxÞ
(73)
Introducing the specific impedance (74a) as the impedance divided by that of an acoustic
plane wave (74a), the impedance boundary condition at the lower wall (73) becomes (74b)
Z ðxÞ
:
z ðxÞ q0 c0
M0
iX
LP0 ð0; k; xÞ ¼ P ð0; k; xÞ
1
z ðxÞ
z ðxÞ
(74a,b)
A similar impedance boundary condition (75b) applies at the upper wall
zþ ðxÞ Zþ ðxÞ
:
q0 c0
1þ
M0
iX
LP0þ ðL; k; xÞ ¼
Pþ ðL; k; xÞ
zþ ðxÞ
zþ ðxÞ
(75a,b)
where the wall specific impedance may be different (75a). The difference in signs between the
impedance boundary conditions at the lower (74b) and upper (75b) walls where the outward
unit normals are opposite is due to the uniform cross flow velocity causing an outflow in the
former and an inflow in the latter case.
Matching conditions of the acoustic pressure at the edges of the boundary layers
The acoustic pressure spectrum P in equation (58) and horizontal (70a) and vertical (70b)
velocity perturbations must be continuous at the edge of the boundary layer and core flow;
this is equivalent42,44 to the continuity of the wave pressure and its first two derivatives.
These three conditions are applied at the junction of the lower (76a-c) and upper (77a-c)
boundary layers with the core flow (67a-b), respectively, at y ¼ L and y ¼ L Lþ
18
International Journal of Aeroacoustics 0(0)
P ðL Þ ¼ P1 ðL Þ; LP0 ðL Þ ¼ iuþ Cþ eijþ L iu C eij L
L2 P00 ðL Þ ¼ u2þ Cþ eijþ L u2 C eij L
(76a-c)
Pþ ðL Lþ Þ ¼ P1 ðL Lþ Þ; LP0þ ðL Lþ Þ ¼ iuþ Cþ eijþ ðLLþ Þ iu C eij ðLLþ Þ
L2 P00þ ðL Lþ Þ ¼ u2þ Cþ eijþ ðLLþ Þ u2 C eij ðLLþ Þ
(77a-c)
where the exponential phase terms involving k and x were omitted and u are the dimensionless vertical upward and downward wavenumbers (68b) in the core flow
u ¼j L¼X
M1 cosh 1 sgnðM0 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ð1 M1 coshÞ2 1 M20 cos2 h
1 M20
(78)
that are real for vertically propagating and imaginary for vertically evanescent or diverging
waves. The wavefield in the duct involves eight unknown quantities: (i) the three coefficients
ðA1 ; A2 ; A3 Þ in equations (64a) and (64b) the lower boundary layer; (ii) likewise ðB1 ; B2 ; B3 Þ
in the upper boundary layer (65a,b); (iii) the amplitudes C of the upward and downward
propagation waves in the core flow (67a) and (67b). These satisfy eight conditions: (a)
matching of the wave pressure and its first two derivatives at the edge of the lower boundary
layer (76a-c); (b) likewise at the edge of the upper boundary layer (77a–c); (c) the two
impedance wall boundary conditions (74a,b;75a,b). Substituting the acoustic pressure
involving the eight coefficients (i–iii) into the eight conditions (a–c) leads to a linear homogeneous system of eight equations for the eight unknowns
½M b ¼ 0
(79)
involving the vector of coefficients b ¼ ½A1 ; A2 ; A3 ; B1 ; B2 ; B3 ; Cþ ; C and the matrix M
where the determinant (80) must be zero for a non-trivial solution
2
Q2 j 1
Q3 j 1
0
0
0
eiuþ d
Q1 j
6 L 1
L 0
L 0
0
6
0
0
0
iuþ eiuþ d
6 L Q1 j 1 L Q2 j 1 L Q3 j 1
6 6 L2
2
2
6 Q00 j L Q00 j L Q00 j
0
0
0
u2þ eiuþ d
6 2 11 2 21 2 31
L
L
6 L
6
6
0
0
0
Q1 j 1
Q2 j 1
Q3 j 1
eiuþ dþ
6
L 0
L 0
L 0
6
0
0
0
Q1 j1 Q2 j1 Q3 j1 iuþ eiuþ dþ
6
6
Lþ
Lþ
Lþ
6
L2 00
L2 00
L2 00
6
6
0
0
0
Q 1 j1
Q 2 j1
Q 3 j1
u2þ eiuþ dþ
2
2
2
6
L
L
L
þ
þ
þ
6
4 #
#
#
0
0
0
0
1
2
3
#þ
#þ
0
0
0
0
#þ
1
2
3
¼0
eiu d
3
7
iu eiu d 7
7
7
7
2 iu d 7
u e
7
7
7
eiu dþ 7
7
7
iu eiu dþ 7
7
7
þ 7
u2 eiu d 7
7
7
5
0
0
(80)
Campos and Legendre
19
where j1 means the evaluation of the function at unity and: (i) the upward and downward
propagating waves in the core flow are multiplied by factors depending on the thickness of
the duct L and of the lower L and upper Lþ boundary layers
d ¼
L
Lþ
; dþ ¼ 1 L
L
(81a,b)
(ii) the impedance boundary condition at the lower (74a,b) and upper (75a,b) impedance
wall involves (82b)
n ¼ 1; 2; 3 :
#n ¼ 1
M0 7L 0
iX
Qn ð0Þ
Qn ð0Þ7
z ðxÞ L
z ðxÞ
(82a,b)
for each of the functions ðQ1 ; Q2 ; Q3 Þ in (64a,b;65a,b).
Eigenvalues as the roots of an 8 8 matrix determinant
The vanishing of the 8 8 matrix determinant (80) specifies the dispersion relation, whose
roots are the eigenvalues for the horizontal compactness (61d), and substituting these in the
wave pressure spectrum leads to the corresponding eigenfunctions. Fixing the core (61a) and
cross (61b) flow Mach numbers, the dimensionless frequency (61c) and the wall specific
impedances (74a, 75a), the determinant (80) depends only on the horizontal compactness
(61d) that: (i) is conserved in the core flow and across to both the upper and lower boundary
layers; (ii) specifies the dimensionless upward and downward vertical wavenumbers (78) that
exist only in the core flow (83a) where holds the dispersion relation (83b)
L y L Lþ :
u2 þ e2 ¼ ðX eM1 uM0 Þ2
(83a,b)
that is the dimensionless form (61a-d) of equation (84)
ðx iU1 k iV0 jÞ2 ¼ c20 ðj2 þ k2 Þ
(84)
arising from the convected wave equation (66). The roots of the determinant (80) may be
complex (85a) so that: (i) the real part of the wavenumber specifies the wavelength k ¼
2p=<ðkÞ and the direction of the propagation in the core flow if jcoshj ¼ jx=½c0 <ðkÞ j < 1;
(ii) the imaginary part specifies the decay =ðkÞ > 0 or growth =ðkÞ < 0 of the, respectively,
stable and unstable modes in the direction parallel to the wall
k ¼ kr þ iki <ðkÞ þ i=ðkÞ
eikx ¼ eikr x eki x
(85a,b)
The corresponding vertical wavenumbers (68b) that exist in the core flow (83a) are also
generally complex with the real part jr specifying the vertical wavelength and the imaginary
part decay ji > 0 or growth ji < 0 in the vertical direction. The exponential growth or decay
is limited in the vertical direction in the core flow (83a) but not in the horizontal direction
1 < x < þ1, because: (i) the horizontal wave number (85a,b) applies across the whole
20
International Journal of Aeroacoustics 0(0)
duct including the core flow and upper and lower boundary layers, and is continuous at the
two interfaces; (ii) the vertical wavenumbers (68b), (78), (83b) and (84) exist only for transversal sinusoidal waves (67b) in the core flow (67a)(68a), and not in the boundary layers
where the waves are sinusoidal (58;70a,b) only on time and longitudinal coordinate but not
in the transverse coordinate.
Eigenfunctions for a duct with lined walls and boundary layers
The preceding theory is illustrated by a set of 24 nozzle flow configurations implying each
case the calculation of eigenvalues and the corresponding eigenfunctions, as illustrated in
the baseline case.
Selection of 24 test cases for nozzle flow configuration
The present model of a lined duct with both shear and cross flow relies on the following
assumptions: (i) boundary layers of thickness L in the upper and lower wall with linear
shear velocity profiles matched to a uniform stream of velocity U1 ; (ii) a cross flow with
uniform velocity V0; (iii) homentropic flow and with constant pressure so that the mean flow
parameters like the speed of sound c0 are constant; and (iv) locally reacting liners in the
upper and lower walls with specific wall impedances (74a,75a). The baseline parameters are
taken as: (i) the speed of sound (86a) in sea level atmospheric conditions; (ii) (iii) a high
subsonic core stream (86b) with a much smaller bias velocity (86c); (iv) boundary layers of
moderate thickness (86d) L ¼ Lm both equal to the thickness of the core flow; (v) a wave
frequency of 108 Hz in the part of audible range 20 Hz 20 kHz (86e); (vi) wall impedances
(86f) or inverse of wall admittance combining resistance and inductance
L ¼ Lm ¼ Lr ¼ 0:5 m;
fc0 ; U1 ; V0 g ¼ f340; 272; 20:4g m=s;
x
¼ 108 Hz;
z ¼1þi
2p
(86a-f)
where Lr is the reference length. These are typical values as order of magnitude of conditions
in the inlet and exhaust ducts of modern jet engines, where the acoustic liners with bias flow
are used to attenuate the sound; for example, the precise values of the specific admittance
depend on frequency and may not be available for a specific application, but it is known that
the real and imaginary parts do not usually exceed unity. The preceding baseline values
(86a-f) serve only to calculate the reference values of the four dimensionless parameters,
namely the specific impedance (86f) and: (i)(ii) the core (87a) and cross flow (87b) Mach
numbers; (iii) the dimensionless frequency(87c)
M1 U1
¼ 0:8;
c0
M0 V0
¼ 0:06;
c0
X xL
¼1
c0
(87a-c)
These reference dimensionless values (87a-c) will each be varied in turn over a sufficiently
wide range to cover most aeronautical applications, so that the choice of starting numbers
(87a-c) is not critical.
Campos and Legendre
21
Computation of eigenvalues as the roots of an 8 8 matrix determinant
The first eight eigenvalues were computed as the smallest complex roots in modulus of 8 8
matrix determinant (80). The values of the determinant were calculated within a 20 20
square for the real (88b) and imaginary (88c) parts of the horizontal wave number
kr þ iki kn ;
10 <ðkÞ; =ðkÞ 10
(88a-c)
The method used for finding all the roots without skipping any of them combines45,46 the
generation of random numbers (Monte-Carlo) for distributing the values to be computed to
the real-imaginary plane and Delaunay’s triangulation for verifying whether or not the value
sampled in the real-imaginary plane is a root of the determinant (80) among its neighbours.
The steps of the method are described as follows
(i) Using pseudo-random number generator algorithm, N ¼ 1:6 105 points are sampled
inside the domain (88c) and some of the points are forced to be on the boundaries as
depicted in the simplified sketch of Figure 3(a). With this number of points, it is
assumed that: (i) the points are evenly spaced between each other as long as N remains
large; (ii) the maximum distance between any pair of two contiguous points does not
exceed Dk ¼ 0:1. There are many pseudo-random number generators available as opensource algorithms, and the authors used NumPy a python library for the random
sampling and Delaunay’s triangulation.
(ii) Delaunay’s triangulation procedure is applied to such points to generate a mesh as
depicted in Figure 3(b). Due to the connectivity of the mesh, for each point i, the
contiguous points can be determined in a given neighbourhood. In this work, the
(a)
(b)
(c)
(d)
Figure 3. Root finding procedure. (a) Random distribution of points in the real vs. imaginary plane. (b)
Delaunay’s triangulation of points. (c) Neighbourhood of points. (d) Verification of local maxima.
22
International Journal of Aeroacoustics 0(0)
neighbourhood is defined as the nearest points with one degree of connection, as
depicted in Figure 3(c).
(iii) Using the N points and connectivity, the inverse of the determinant (80) is evaluated
and then each point is checked if it is a local maximum. The local maxima kn are
defined as points in which the inverse of the value of the determinant is greater than the
value of its neighbours; an example is depicted in Figure 3(d). The boundary points on
the square domain (88c) are excluded of the cheeking procedure.
(iv) With the first eight local maxima corresponding to the eigenvalues kn, steps (i) to (iii)
are repeated in a polygonal domain centered at each local maxima kn with approximately the same number of points N ¼ 1:6 105 but in a domain much smaller. This
loop is repeated three times in all eigenvalues to reduce errors below O104 .
For the determination of the first eight roots (per configuration), with an accuracy about
104 , the determinant (80) is evaluated 4 106 (25 1:6 105 ) times taking around
2532 s of computation time using a single processor Intel Core i5-2500k at 3.70 GHz.
Without the incremental root finding procedure described above, i.e. using brute-force
searching, to obtain similar precision ( 104 ) in the determination of the eigenvalues, the
number of points to be evaluated should rise up to 1010 . In other words, 2500 times slower
than the method proposed. Although tedious, this approach ensures that no roots are
missed, and the eight chosen eigenvalues are really the smallest in the modulus. The use
of classic numerical methods for finding roots, such as Newton–Raphson, will not give this
assurance since the determinant (80) may not be monotonic or convex. Moreover, the particular choice of using a Delaunay’s triangulation for constructing a tree to find roots is
driven by: (i) the low computational cost for the construction of the triangulation in comparison to the computation of the determinants; (ii) more direct neighbors are allowed than
in a squared mesh. For each eigenvalue, the corresponding eigenfunction is calculated using
(64a,b) and (65a,b) in the lower and upper layers and (67a,b) in the core flow; the respective
amplitudes (A13 ; B13 and C ) are determined by the matching conditions (76a-c;77a-c) at
the edges of the boundary layers and at the wall impedance boundary conditions (74a,b;75a,
b). Since the determinant (80) is zero, the amplitudes are determined to within a multiplying
constant, specified by normalizing the acoustic pressure perturbation to the value at the wall
that the flow comes out of, i.e. the lower wall for M0 > 0 and the upper wall for M0 < 0. The
first eight eigenvalues have been computed (Tables 1 to 5) and the corresponding
Table 1. First eight eigenvalues of the horizontal wavenumber k determined by the roots of the determinant (80) with smallest modulus for four values of the cross Mach number (91a-d).
kn
(91a)
(91b)
(91c)
(91d)
1
2
3
4
5
6
7
8
Figures
0.2454 0.3977i
0.4651 þ 0.3338i
0.6665 þ 0.1778i
1.7018 0.8771i
1.8228 þ 0.7194i
2.0924 0.2945i
2.5 þ 0.0i
2.8473 1.0622i
5a-d
1.1111 þ 0.0i
1.1069 0.3539i
0.1423 þ 1.388i
0.0252 þ 1.4451i
1.3888 þ 0.6445i
0.1132 1.5624i
0.2778 1.6399i
1.5468 0.9838i
6a-d
0.1092 0.0508i
1.0277 þ 0.1477i
2.5 þ 0.0i
2.6471 0.5116i
2.8796 þ 0.3463i
2.91 þ 0.895i
3.0265 0.5242i
4.9762 þ 0.0476i
7a-d
0.6361 þ 0.1759i
0.6942 0.2455i
0.8001 þ 0.0226i
1.062 0.78i
1.1341 þ 0.7172i
0.3148 1.7228i
0.2914 þ 1.7819i
2.5 þ 0.0i
8a-d
Campos and Legendre
23
eigenfunctions plotted (Figures 4 to 27) in five sets of several combination of parameters,
from the baseline case (87a-c;86f) giving each parameter in turn a relatively wide range of
five values. Since the eigenfunctions are complex, the amplitudes and phases are plotted
separately versus distance from the wall divided by the reference length Lr as discussed next.
Table 2. First eight eigenvalues of the horizontal wavenumber k determined by the roots of the determinant (80) with smallest modulus for four different values of the core Mach number (92a-d).
kn
(92a)
(92b)
(92c)
(92d)
1
2
3
4
5
6
7
8
Figures
1.0 0.3422i
0.999 þ 0.3453i
3.1371 0.0202i
3.1419 þ 0.005i
3.177 0.2957i
3.1813 þ 0.2804i
1.7823 þ 6.9811i
1.7352 7.0145i
9a-d
0.3571 þ 0.5146i
0.6712 0.4522i
0.7879 0.205i
2.2232 þ 0.0916i
2.1991 þ 1.4239i
2.3499 1.3344i
2.5204 0.9838i
2.6392 þ 1.161i
10a-d
0.2456 þ 0.3984i
0.465 0.3332i
0.6667 0.1764i
1.7084 þ 0.8825i
1.8153 0.713i
2.059 0.4026i
2.0964 þ 0.2946i
2.5 þ 0.0i
11a-d
0.1532 0.2703i
0.2937 þ 0.2213i
0.4945 þ 0.1367i
1.1176 0.502i
1.188 þ 0.4232i
1.3067 þ 0.2737i
1.3507 0.2678i
1.6667 þ 0.0i
12a-d
Table 3. First eight eigenvalues of the horizontal wavenumber k determined by the roots of the determinant (80) with smallest modulus for five different values of specific impedance (93a-e).
kn
(93a)
(93b)
(93c)
(93d)
(93e)
1
2
3
4
5
6
7
8
Figures
0.3025 0.1181i
0.3029 þ 0.1186i
0.4769 þ 0.0007i
1.8438 0.8804i
1.8503 þ 0.8866i
2.1874 0.3296i
2.193 þ 0.3312i
2.5 þ 0.0i
13a-d
0.1015 0.4497i
0.5351 þ 0.3202i
0.5992 þ 0.2997i
1.7846 þ 0.4944i
1.7025 0.992i
2.0656 0.2652i
2.0463 þ 0.5489i
2.5 þ 0.0i
14a-d
0.3775 0.4301i
0.3777 þ 0.4307i
0.8696 þ 0.001i
1.6882 0.7413i
1.6959 þ 0.747i
2.0509 0.3115i
2.0546 þ 0.312i
2.5 þ 0.0i
15a-d
0.2313 0.3492i
0.2316 þ 0.3501i
0.7344 þ 0.0003i
2.0204 0.0017i
1.9088 1.0765i
2.5 þ 0.0i
2.4642 þ 0.8771i
2.4642 0.8776i
16a-d
0.0367 0.3354i
0.377 þ 0.2613i
0.6297 þ 0.1461i
1.6174 þ 0.1714i
1.8081 1.0139i
2.0896 0.1699i
2.2493 þ 0.6199i
2.5 þ 0.0i
17a-d
Table 4. First eight eigenvalues of the horizontal wavenumber k determined by the roots of the determinant (80) with smallest modulus for different values of specific impedance at the upper wall (94b-e), with
specific impedance at the lower wall (94a).
kn
(94b)
(94c)
(94d)
(94e)
1
2
3
4
5
6
7
8
Figures
0.2878 þ 0.3922i
0.3983 0.392i
0.6676 0.0247i
1.7697 þ 0.7845i
1.7795 0.8388i
1.9735 þ 0.4708i
2.2071 0.493i
2.5 þ 0.0i
18a-d
0.0975 0.2673i
0.4685 þ 0.2396i
0.5488 þ 0.1152i
1.7228 0.93i
1.9456 þ 0.2754i
1.8761 þ 0.7817i
2.3045 0.0877i
2.5 þ 0.0i
19a-d
0.2742 0.5019i
0.473 þ 0.4737i
0.8433 þ 0.1718i
1.6496 0.8339i
1.7433 þ 0.6536i
1.9674 0.4103i
2.0994 þ 0.4995i
2.5 þ 0.0i
20a-d
0.0638 þ 0.4818i
0.5345 0.2883i
0.6145 0.3731i
1.8518 þ 0.5026i
1.8589 0.9681i
2.5 þ 0.0i
2.4064 0.8872i
2.9083 0.5133i
21a-d
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International Journal of Aeroacoustics 0(0)
Table 5. First eight eigenvalues of the horizontal wavenumber k determined by the roots of the determinant (80) with smallest modulus for different values of proportions of upper, lower and central layers
(95a-f).
kn
(95a)
(95b)
(95c)
1
2
3
4
5
6
7
8
Figures
0.3983 þ 0.6634i
0.757 þ 0.3206i
1.0164 1.0329i
1.6531 þ 0.7341i
1.6909 þ 0.6681i
1.787 1.5218i
2.5 þ 0.0i
3.5438 1.0444i
22ad
0.8104 þ 0.3616i
0.3477 þ 0.8888i
1.4736 þ 0.7885i
1.6034 þ 0.7949i
0.2016 1.9742i
2.5 þ 0.0i
2.4733 þ 1.3711i
3.0247 0.2405i
23ad
0.8635 0.2069i
0.4135 0.921i
1.4781 0.8176i
1.5835 0.9026i
2.5 þ 0.0i
2.2206 1.5739i
2.9404 þ 0.5187i
3.4326 þ 0.9475i
24ad
(95d)
(95e)
(95f)
0.7685 0.1447i
0.4106 0.6829i
0.9573 þ 1.0388i
1.5695 0.7652i
1.8058 0.7236i
1.8345 þ 1.4959i
2.5 þ 0.0i
2.8788 þ 0.4875i
25a-d
0.7414 þ 0.1353i
0.7264 þ 1.1239i
0.1977 2.0026i
2.5 þ 0.0i
2.1673 þ 1.6927i
2.9112 1.4096i
3.9362 þ 0.7985i
3.70631.7653i
26a-d
0.767 0.0556i
0.7759 1.1789i
0.2286 þ 2.0032i
1.6836 1.5076i
2.5 þ 0.0i
2.0439 1.903i
2.8656 þ 1.127i
3.6505 þ 2.0305i
27a-d
1
2
3
4
5
6
7
8
Figures
Amplitude and phase of the eigenfunctions for the baseline case
The 23 test cases for nozzle flow configurations are variations (Figures 5 to 27) around the
baseline case (Figure 4). In each test case are considered: (i) the eight eigenvalues with the
smallest modulus, for the five sets of variations of parameters (Tables 1 to 5) around the
baseline case (87a) to (87c); (86f); (ii) the corresponding eight eigenfunctions plotted separately as amplitudes and phase (Figures 4 to 27), with overlap leading to only four curves in
one case (Figure 9). This particular case is in the absence of core flow (89a) leading by (68b)
to vertical wavenumbers (89b)
M1 ¼ 0 :
j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x=c0 sgnðM0 Þ ðx=c0 Þ2 k2 1 M20
1 M20
(89a,b)
that depend only on k2, so that the modes corresponding to k overlap. Note also that
changing the sign of the cross-flow Mach number, that is from upflow to downflow, in
general (89b) interchanges the vertical wavenumbers, so the eigenfunctions are mirror
images relatively to the axis; this can be confirmed by comparing the baseline case
(Figure 4) with the change of cross flow Mach numbers from baseline (87b) to M0 ¼
0:06 (Figure 5). The baseline case is considered in more detail next as a preliminary to
a more cursory examination of the other 22 test cases (Figures 6 to 27) that are variations
around it.
Campos and Legendre
25
(a)
(b)
(c)
(d)
Figure 4. Amplitude (a,b) and phase (c,d) of the first eight eigenfunctions using the baseline case parameters (87a–c) with the same specific impedance value at the lower and upper walls (86f). The amplitude has
been normalized by the acoustic pressure perturbation to the value at the wall that the flow comes out of, in
this case the lower wall. The shaded strip represents the middle layer with uniform velocity. The corresponding eigenvalues are (90a–h).
The first eight eigenvalues of the horizontal wavenumber k as the roots of the determinant (80) with the smallest modulus are computed for the baseline case (87a-c;86f)
k1 ¼ 0:2454 0:3977i; k2 ¼ 0:4651 þ 0:3338i; k3 ¼ 0:6665 þ 0:1778i
k4 ¼ 1:7018 0:8771i; k5 ¼ 1:8228 þ 0:7194i; k6 ¼ 2:0924 0:2945
k7 ¼ 2:5 þ 0:0i; k8 ¼ 2:8473 1:0622i
(90a-h)
The base line case (Figure 4(a) to (d)) corresponding to an upward cross flow with small
Mach number (87b, M0 ¼ 0:06) and equal wall impedance in the upper and lower walls
consisting of a unitary positive resistance and inductance (z ¼ 1 þ 1i, 86f) leads to: (i) an
amplitude minima near the axis of the duct for the modes 1 to 3, monotonic decrease and
increase away from the lower wall for the modes 4 and 5, respectively, and an amplitude
oscillation for the modes 6 to 8; (ii) the amplitude variation is large (small) for the modes
26
International Journal of Aeroacoustics 0(0)
(a)
(b)
(c)
(d)
Figure 5. As Figure 4 with a downward cross flow Mach number (91a). The corresponding eigenvalues can
be found in Table 1, first column.
1 to 5 (6 to 8), and is monotonic (non-monotonic) for the modes 4 and 5 (all other); (iii) the
phase reduces (increases) away from the lower wall for the modes 2, 3 and 6 (for the mode 1)
with a rapid change in the core flow, for the modes 4 and 7 increases slowly away from the
wall, and for the mode 5 (mode 8) increases (remains small) with a rapid change at the lower
(upper) boundary layers.
23 Variations of parameters around the baseline case
The eight roots of the determinant (80) with the smallest modulus specify the lowest-order
eigenvalues of the acoustic pressure perturbation in a parallel-sided duct with impedance
walls, uniform cross-flow and boundary layers with linear shear flow profile and the corresponding complex eigenfunctions are plotted (Figures 5 to 27) in five sets of several combinations of parameters illustrating the dependence on: (i–ii) the cross and core flow Mach
numbers; (iii–iv) equal and different wall specific impedances at the upper and lower walls;
and (v) relative thickness of the upper and lower boundary layers.
Campos and Legendre
27
(a)
(b)
(c)
(d)
Figure 6. As Figure 4 with absent cross flow (91b). The corresponding eigenvalues can be found in Table 1,
second column.
(a)
(b)
(c)
(d)
Figure 7. As Figure 4 with an upward cross flow Mach number (91c). The corresponding eigenvalues can
be found in Table 1, third column.
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International Journal of Aeroacoustics 0(0)
(a)
(b)
(c)
(d)
Figure 8. As Figure 4 with a larger upward cross flow Mach number (91d). The corresponding eigenvalues
can be found in Table 1, fourth column.
(a)
(c)
(b)
(d)
Figure 9. As Figure 4 without core flow Mach number (92a). The corresponding eigenvalues can be found
in Table 2, first column.
Campos and Legendre
29
(a)
(c)
(b)
(d)
Figure 10. As Figure 4 with a low subsonic upstream core flow Mach number(92b). The corresponding
eigenvalues can be found in Table 2, second column.
(a)
(b)
(c)
(d)
Figure 11. As Figure 4 with a high subsonic upstream core flow Mach number (92c). The corresponding
eigenvalues can be found in Table 2, third column.
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International Journal of Aeroacoustics 0(0)
(a)
(b)
(c)
(d)
Figure 12. As Figure 4 with a low supersonic downstream core flow Mach number (92d). The corresponding eigenvalues can be found in Table 2, fourth column.
(a)
(b)
(c)
(d)
Figure 13. As Figure 4 with rigid lower and upper walls (93a). The corresponding eigenvalues can be found
in Table 3, first column.
Campos and Legendre
31
(a)
(b)
(c)
(d)
Figure 14. As Figure 4 with resistive specific impedance at the lower and upper walls (93b). The corresponding eigenvalues can be found in Table 3, second column.
(a)
(b)
(c)
(d)
Figure 15. As Figure 4 with equal positive inductive specific impedance at the lower and upper walls (93b).
The corresponding eigenvalues can be found in Table 3, third column.
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International Journal of Aeroacoustics 0(0)
(a)
(b)
(c)
(d)
Figure 16. As Figure 4 with equal negative inductive specific impedances at the lower and upper walls
(93d). The corresponding eigenvalues can be found in Table 3, fourth column.
(a)
(c)
(b)
(d)
Figure 17. As Figure 4 with complex conjugate mixed resistive-inductive specific impedance at the lower
and upper walls (93e). The corresponding eigenvalues can be found in Table 3, fifth column.
Campos and Legendre
33
(a)
(b)
(c)
(d)
Figure 18. As Figure 4 with rigid upper wall (94b). The corresponding eigenvalues can be found in Table 4,
first column.
(a)
(b)
(c)
(d)
Figure 19. As Figure 4 with resistive specific impedance at the upper wall (94c). The corresponding
eigenvalues can be found in Table 4, second column.
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International Journal of Aeroacoustics 0(0)
(a)
(b)
(c)
(d)
Figure 20. As Figure 4 with positive inductive specific impedance at the upper wall (94d). The corresponding eigenvalues can be found in Table 4, third column.
(a)
(b)
(c)
(d)
Figure 21. As Figure 4 with negative inductive specific impedance at the upper wall (94e). The corresponding eigenvalues can be found in Table 4, fourth column.
Campos and Legendre
35
(a)
(b)
(c)
(d)
Figure 22. As Figure 4 with thicknesses of the upper and lower boundary layers and the core flow in the
proportions (95a). The corresponding eigenvalues are in Table 5, first column.
(a)
(c)
(b)
(d)
Figure 23. As Figure 4 with the 3 flow layers in the proportions (95b). The corresponding eigenvalues are
in Table 5, second column.
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International Journal of Aeroacoustics 0(0)
(a)
(c)
(b)
(d)
Figure 24. As Figure 4 with the triple deck in the proportions (95c). The corresponding eigenvalues are in
Table 5, third column.
(a)
(b)
(c)
(d)
Figure 25. As Figure 4 with the proportion (95d) for the triple flow layers. The corresponding eigenvalues
are in Table 5, fourth column.
Campos and Legendre
37
(a)
(b)
(c)
(d)
Figure 26. As Figure 4 with a proportion (95e) for the triple deck. The corresponding eigenvalues are in
Table 5, fifth column.
(a)
(c)
(b)
(d)
Figure 27. As Figure 4 with a proportion (95f) for the thicknesses of the upper and lower boundary layers
and core flow. The corresponding eigenvalues can be found in Table 5, sixth column.
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International Journal of Aeroacoustics 0(0)
Effect of longitudinal core and transverse bias flow Mach numbers
A set of four values of the cross flow Mach number was selected for the computation of the
duct modes
M0 ¼ 0:06; 0; 0:3; 0:8;
(91a-d)
The values correspond to upward (positive), downward (negative) and absent cross flow
Mach numbers.
The first eight eigenvalues of the horizontal wavenumber k as the roots of the determinant (80) with the smallest modulus are computed and listed in Table 1 with the corresponding eigenfunctions plotted in Figures 5 to 8.
The case (86f;87a,c) without cross flow (91b) is the simplest for which the wave equation
reduces to Pridmore-Brown’s form.10 In this case (Figure 6(a) to (d)), the waveforms are
symmetric relative to the duct axis because the impedance (86f) is the same at the two walls.
The amplitude (Figure 6(a) and (b)) is nearly uniform in the core flow and varies mostly in
the boundary layers for the modes 1 and 2; the phase (Figure 6(c) and (d)) is almost constant
for both modes but larger for the mode 2 than for the mode 1. In contrast, the modes 3 and 4
have significant amplitude variation in the core flow with maximum on the axis, where the
phase is minimum. The modes 5 and 8 have small variations in amplitude with a minimum
in the core flow, while their phase varies significantly with also a minimum in the core flow
for the mode 5 and a maximum for the mode 8. The modes 6 and 7 have variations in
amplitude and phase with a maximum in the core flow. The amplitude changes are larger for
the modes 3, 4, 6 and 7 that have a maxima; the phase changes are larger for the modes 6 to
8 than for the modes 3 to 5. Figure 5(a) to (d) concerns the baseline case (87a-c;86f) with
reverse cross flow (91a) instead of (87b); the eigenfunctions in Figure 5(a) to (d) are mirror
images relative to the duct axis of those in Figure 4 because all parameters other than the
cross flow Mach number are equal and the acoustic pressure perturbation is normalized to
the value at the wall with outflow, that is lower (upper) wall in Figure 4(a) to (d) (Figure 5(a)
to (d)).
This confirms that the change of sign of the cross flow from upward to downward: (i)
does not change the eigenvalues that coincide in (90a-h) and the first column of Table 1; (ii)
changes the eigenfunctions to mirror images, that is anti-symmetric relative to the duct axis
as shown in Figures 4 and 5. Thus, there is no need to consider negative cross flow Mach
numbers and only two more positive values (91c,d) are considered for the eigenvalues
(columns 3 and 4 of Table 1) and eigenfunctions (Figures 7 and 8). Increasing the crossflow Mach number to a low subsonic value (91c) (Figure 7(a) to (d)) shows that: (i) for the
modes 1 and 2, the amplitude varies smoothly with (without) a maximum value in the core
flow for the mode 1 (mode 2) and the phase in both modes monotonically increases from the
lower to the upper wall; (ii) the modes 3 and 4 present smaller variations when compared to
the modes 1 and 2, with amplitudes decreasing from the lower to upper layers and phases
with almost no variation; and (iii) the modes 5, 7 and 8 present sinusoidal-like oscillations in
amplitude with smooth phase transition in the core flow for the modes 5 and 7 and strong
phase changes across all the three layers for the mode 8; (iv) the mode 6 smoothly increases
in amplitude from the lower to the upper wall with an almost linear increase in phase; and
(v) the amplitude and phase variations are larger for the modes 1, 2 and 6. Increasing the
cross-flow Mach number to high subsonic (91d) leads (Figure 8(a) to (d)) to: (i) largest
Campos and Legendre
39
amplitude variation for the modes 1 and 8, that exhibit large amplitude oscillations; (ii)
other modes have smaller monotonic or oscillatory amplitudes; (iii) the phase changes, again
monotonic for some and oscillatory for other modes, are not large.
A set of four values of the core flow Mach number distinct from the baseline case (87a)
was selected for the computation of the duct modes. The values correspond to downstream
(positive), upstream (negative) and zero core flow Mach numbers
M1 ¼ 0; 0:6; 0:8; 1:2
(92a-d)
ranging from subsonic to transonic and supersonic regime. The first eight eigenvalues of the
horizontal wavenumber k as the roots of the determinant (80) with the smallest modulus are
listed in Table 2 with the corresponding eigenfunctions plotted in Figures 9 to 12.
The case (86f;87b,c) without shear flow (92a) leads to a convected wave equation due only
to transverse bias flow, implying that the eigenfunctions are not symmetric relatively to the
axis (Figure 9(a) to (d)) and the eigenvalues came in pairs with almost the same amplitude
but opposite sign, i.e. k1 ¼ k2 ; k3 ¼ k4 ; k5 ¼ k6 ; k7 ¼ k8 , as predicted before (89a)
because the vertical wavenumbers in the core flow (89b) are independent of the sign of the
horizontal wavenumber k; (ii) confirmed by the values in the column 1 of Table 2. Therefore,
the modes 1,2; 3,4; 5,6 and 7,8 are nearly superimposed in Figure 9(a) and (b) in which: (i)
the mode 12 has a minimum amplitude in the core flow decreasing (increasing) towards the
lower (upper) wall with constant but different phase in the lower and upper boundary layers
having a strong transition in the middle; (ii) the mode 34 has a maximum amplitude in the
core flow monotonically increasing (decreasing) in the lower (upper) boundary layers, while
the phase is nearly constant; (iii) the mode 56 also presents a maximum amplitude in the
core flow increasing monotonically in the lower layer and with oscillatory behaviour in the
upper layer with also a nearly constant phase; (v) the mode 78 has strong variations in
phase and amplitude in the lower layer, a maximum amplitude in the core flow and the
amplitude decreases with oscillations in the upper layer, while the phase increases monotonically except for the oscillation in the lower layer; and (vi) the amplitude variations are
large for all the modes and only the modes 3 to 6 have small phase changes. The eight modes
are distinct either for positive (baseline, Figure 4(a) to (d)) or negative (Figure 10(a) to (d))
core flow Mach number.
The change in direction and magnitude from equation (87a) in Figure 4(a) to (d) to
equation (92b) in Figure 10(a) to (d), with other parameters (86f;87b,c) unchanged, affects
significantly only some modes: (i) the mode 4 changes from monotonic increasing to an
amplitude with a minimum in the core flow, and from increasing to decreasing phase; (ii)
there less difference between the lower modes 1 to 3 in the two cases of Figures 4(a) and b
and 10(a) and (b); (iii) the large amplitude and phase changes of the mode 5 in Figure 4(a) to
(d) also apply to the mode 7 in Figure 10(a) to (d). The direct comparison of the baseline
case (86f;87a-c) with the same core flow Mach number (Figure 4(a) to (d)) reversed (Figure
11(a) to (d)) shows that a mean flow reversal does not change too much the amplitudes and
phases of the lowest order eight modes. A higher low supersonic (92d) core flow Mach
number (Figure 12(a) to (d)) compared with the high subsonic (87a) core flow Mach
number with the same direction, and other parameters identical (86f;87b,c) to the baseline
case (Figure 4(a) to (d)) shows: (i) small differences in amplitude and phase for lower modes
1 to 4; (ii) more significant changes for higher modes 5 to 8, for instance the modes 5 to 7
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International Journal of Aeroacoustics 0(0)
have more pronounced amplitude minima in the core flow in Figure 12(b) compared with
Figure 4(b), and the phase changes are different in Figures 4(d) and 12(d).
Effect of equal and different wall impedances on the normal modes
A set of five values of the specific wall impedance of the liners distinct from the baseline case
(86f) was selected for the computation of the duct modes, starting with both walls, upper
and lower having the same value of impedance. The specific impedance of a locally acoustic
liner depends on the frequency and the real and imaginary parts usually do not exceed unity.
The specific impedance is given by the values
z ¼ 1; 1; þi; i; 1 i;
(93a-e)
including rigid wall, resistive, inductive and combined impedances. The first eight eigenvalues of the horizontal wavenumber k as the roots of the determinant (80) with the smallest
modulus are listed in Table 3 and the corresponding eigenfunctions are plotted in Figures 13
to 17.
The baseline case (87a-c) with rigid walls (93a) leads (Figure 13(a) to (d)) to: (i) a minimum amplitude for the modes 1, 2 and 3 in the core flow increasing monotonically towards
the walls, monotonic decrease and increase away from the wall respectively for the modes 4
and 5, and amplitude oscillations for modes 6 to 8; (ii) the phase reduces away from the
lower wall for the modes 1–3, 6 and 7 with a rapid change in the core flow for the modes 1
and 2, and increases monotonically away from the lower wall for the modes 4, 5 and 8, the
last one with small variation. It could be expected that the larger flow velocity in the core
flow would lead to larger phase changes, as is visible for the modes 1–3 and 5–7, but this is
not always the case as shown by the modes 4 and 8. The case of a less rigid but still resistive
impedance (93b) in Figure 14(a) to (d): (i) changes most the amplitude and phase of the
modes 1 and 4 relative to the rigid wall in Figure 13(a) to (d); (ii) the modes 5 to 8 are also
modified more for the amplitudes that tend to oscillate more than for the phases. A purely
inductive wall impedance (93c) in Figure 15(a) to (d) relative to the rigid wall in Figure 13(a)
to (d) changes most the phases of the modes 1 and 2 from decreasing to increasing away
from the lower wall. The opposite inductive wall (93d) in Figure 16(a) to (d) changes all the
modes relative to the rigid wall in Figure 13(a) to (d), most noticeably the amplitude and
phase of the modes 4–6 and 8. A combined resistive-inductive impedance (93d) in Figure 17
(a) to (d) preserves best the amplitudes and phases of the modes 1 and 3 relative to the rigid
walls in Figure 13(a) to (d) and changes most the phase and amplitude of mode 4.
In addition to the preceding cases, a set of four values corresponding to different specific
wall impedances at the upper and lower walls was selected for the computation of the duct
modes. For the lower wall, the impedance is kept at the value of the baseline case (86f)
(94a) and for the upper wall, the specific impedances are given by the values (94b-e)
z ¼ 1 þ i :
zþ ¼ 1;
1; þ i; i
(94a-e)
The first eight eigenvalues of the horizontal wavenumber k as the roots of the determinant (80) with the smallest modulus are listed in Table 4 and the corresponding eigenfunctions are plotted in Figures 18 to 21.
Campos and Legendre
41
Keeping the specific impedance of the lower (86f)(94a) and changing the specific impedance of the upper wall from equal to the lower wall (86f) to rigid wall (94b) in the baseline
case (87a-c) modifies the acoustic field from Figures 4(a) to (d) to 18(a) to (d), most noticeably: (i) the amplitude minima of the modes 1 to 3 are much less pronounced and the mode 4
changes from decreasing to increasing amplitude away from the lower wall; (ii) the phase
changes most for the mode 1 from increasing to decreasing in both cases with a fast change
in the core flow; (iii) the modes 5 to 8 are all significantly affected with more pronounced
amplitude and phase oscillations. Leaving all other parameters (87a-c) and the impedance of
the lower wall (94a) unchanged, and changing the upper wall from rigid (94b) in Figure 18
(a) to (d) to moderately resistive (94c) in Figure 19(a) to (d), the amplitudes and phases of
the first eight modes become closer to the baseline case (86f) in Figure 4(a) to (d), showing
that a rigid wall changes more the sound field than a moderate resistance. Changing the
upper wall from resistive (94c) in Figure 19(a) to (d) to positive inductive (94d) in Figure 20
(a) to (d) affects more the higher modes 5 to 8 than the lower modes 1 to 4, for example the
mode 5 has a more pronounced amplitude minimum and a sharper phase change in the core
flow with overall shift to negative phases. Inverting the inductive impedance at the upper
wall from positive (94d) in Figure 20(a) to (d) to negative (94e) in Figure 21(a) to (d) affects
significantly all the modes, for example: (i) the mode 4 changes in amplitude from monotonic decreasing from the lower to the upper wall to having an amplitude minimum in the
core flow and the phase changes from increasing to decreasing away from the lower wall; (ii)
the mode 5 changes in amplitude from increasing oscillation to monotonic decrease, and the
phase also changes from oscillatory to monotonic, from fast to slow change in the core flow.
Besides dissimilar impedances at the upper and lower walls another type of asymmetry are
different thicknesses of the upper and lower boundary layers, compared to each other and
the thickness of the core flow.
Effects of unsymmetrical wall boundary layers and core flow thickness
A set of six combinations of the proportions of the upper and lower boundary layers relative
to the thickness of the core flow was selected for the computation of the duct modes. The
values corresponding to thicknesses of: (i) the upper boundary layer Lþ ; (ii) the lower
boundary layer L ; (ii) and the middle layer with constant velocity Lm are given scaled to
the reference length Lr. The proportions chosen are
P :¼
Lþ L Lm
;
;
Lr Lr Lr
¼ f1; 0:5; 1:5g;
f1; 0:25; 1:75g;
f0:5; 0:25; 2:25g;
f0:25; 1; 1:75g;
f0:5; 1; 1:5g;
f0:25; 0:5; 2:25g
(95a-f)
The first eight eigenvalues of the horizontal wavenumber k as the roots of (80) with the
smallest modulus are listed in Table 5 and the corresponding eigenfunctions are plotted in
Figures 22 to 27.
In Figures 4 to 21, the shaded area corresponds to the core flow demonstrating its
thickness relative to the lower and upper boundary layers. In Figures 4 to 21, that were
concerned with the effects of other parameters, the three values of the thickness of the core
flow and two boundary layers were kept equal (86d); the purpose of Figures 22 to 27 is to
42
International Journal of Aeroacoustics 0(0)
vary this proportion for the same distance between the walls. Figure 22(a) to (d) concerns
(95a) an upper boundary layer twice the thickness of the lower boundary layer, with their
sum equal to the thickness of the core flow: (i) the amplitude is minimum near the axis for
the modes 1 to 3 and rapidly increases away from the core flow towards the upper wall for
the mode 4, whereas the modes 5 to 8 exhibit amplitude oscillations; (ii) the phases vary
smoothly for all the modes except for modes 3 and 4 that have phase jumps not far from the
axis in opposite direction, respectively, to higher and lower phases; (iii) the phase decreases
away from the lower wall for the modes 2, 4, 5 and 8, increases for the modes 3 and 6, is
almost constant for the mode 7 and is not monotonic with a minimum near the axis for the
mode 1. Figure 23(a) to (d) keeps (95b) the thickness of the upper boundary layer and makes
the lower boundary layer thinner by a quarter and with the core flow seven times thicker.
The thin lower boundary layer has little effect on the waveform with the dominant effects
coming from the thick core flow. The largest effects are on the amplitudes of the modes 4 to
8 and phase of all modes except for the mode 4, for which the effect of the thicker core flow
is to make the phase jump at the middle more gradual. Figure 24(a) to (d) exchanges
equation (95c) relative to Figure 23(a) to (d) the lower for the upper boundary layer thicknesses (95b) while keeping the thickness of the core flow, that thus has a lower boundary
layer four times thicker than the upper boundary layer. Such a difference affects significantly
all the modes. Figure 25(a) to (d) reverses equation (95d) the thickness of the upper and
lower boundary layers relative (95a) to Figure 22(a) to (d); there are significantly differences
mostly in amplitudes of the modes 4 to 7 and phases of all the modes except for the mode 8.
Figure 26(a) to (d) considers equation (95e) a thin lower boundary layer and upper boundary layer also thin but twice as thick, and a dominant core flow nine times thicker that the
lower boundary layer. Comparing with both boundary layers twice as thick in Figure 22(a)
to (d), there are significant differences like more oscillations and different growth and decay
rates within several cases a growth replaced by a decay; the thicker core flow allows more
oscillations, whereas the thicker boundary layers affects strongly the evolution of the
acoustic-vortical field from the lined impedance wall to the interface between them.
Figure 27(a) to (d) interchanges (95f) the thicknesses of the upper and lower boundary
layers relative to Figure 26(a) to (d). There are again significant differences of amplitudes
and phases of the modes, because the cross-flow has a fixed direction and its effect is distinct
when the thicker boundary layer is crossed first or second.
Conclusions
The acoustics of a parallel-sided two-dimensional nozzle has been considered taking into
account three sets of phenomena than can strongly affect the sound amplitude and phase,
namely (Figure 1) the combined effects of: (i) locally reacting lower and upper walls with
possibly distinct impedances; (ii) upper and lower boundary layers with possibly distinct
thicknesses, matching to the same uniform core flow, through unidirectional shear flows
with linear velocity profiles, whose constant vorticities may be different; (iii) the effect of
cross-flow, or “bias flow” in the acoustic liners, that has most often not been considered in
the scientific literature, although it can strongly affect the sound20,21 even for very low bias
flow Mach numbers. Physically, the longitudinal flow over (i) porous liners causes vortex
shedding,24 and the vortices interact with the shear flow parallel to the wall (ii) and are swept
away from the wall by a bias flow even of small Mach number (iii). The present model
allows for the interactions of all these three effects (i) to (iii) in the aeroacoustics of nozzles.
Campos and Legendre
43
The model involves an acoustic triple deck (Figure 2) with: (i) acoustic-vortical waves of
shear type8–25 in the upper and lower boundary layers, each involving three non-plane wave
modes42,44; (ii) these are coupled to the upstream and downstream propagating plane waves
convected by the uniform core flow. The eight wave amplitudes are determined from eight
conditions: (i) two impedance boundary conditions at the lined walls; (ii) two sets of three
matching conditions at the interfaces between the boundary layers and the core flow. This
leads to an 8 8 matrix determinant as the dispersion relation whose roots specify the
eigenvalues and hence the eigenfunctions. The first eight eigenvalues are calculated
(Tables 1 to 5) and the corresponding eigenfunctions are plotted (Figures 4 to 27) for a
wide range of the parameters involved in the determination of the acoustic field, namely: (a)
the core flow Mach number from low subsonic to supersonic; (b) the bias flow Mach number
from very small to low subsonic; (c) rigid, resistive, inductive or mixed wall impedances
either equal or different at the two walls; and (d) equal or distinct thicknesses of the upper
and lower boundary layers relative to each other and to the thickness of the core flow.
The variety of waveforms cannot be explained by the simplistic predictions using geometrical acoustics such as increasing amplitude and diverging rays propagating upstream,
and the reverse propagating downstream, within a shear flow in a duct. The geometrical
acoustics approximation is far from its domain of applicability here because: (i) the wavelengths can be comparable to the lengthscales of variation of the mean flow velocity, and
thus the sound waves are not sinusoidal in the boundary layers; (ii) the coupling of sound
with vorticity implies that the waves in the boundary layer are not purely acoustic but rather
acoustic-shear modes. The implications are that these non-sinusoidal acoustic-shear waves
unlike sound rays can exchange energy with the mean flow. Thus, the sound attenuation by
the wall liners26–41 is coupled to acoustic-shear waves in the boundary layer that are of
second-order as for sound in the absence of cross-flow, that is not considered in most of the
aeroacoustic literature. The inclusion of a cross-flow raises the order of the acoustic wave
equation from two to three, implying the existence of a third “vortical” mode in the boundary layer in addition to the two acoustic waves. This further interaction is quite significant
even for the very low cross-flow Mach numbers typical of bias flows out of the holes of
perforated acoustic liners, explaining why they are widely used. The variety of acoustic
modes that results from the three interactions (i) to (iii) and the choice of four parameters
(a) to (d) can be seen in Figures 4 to 27, that show how much difference can be made by
changing just one parameter.
The main aim of the present paper on acoustic-vortical waves in a parallel-sided duct is to
model the combined effects of: (a) wall with possibly distinct impedances of the locally
reacting liners; (b) two possibly different boundary layers with linear velocity profile matching to the same free stream; (c) a uniform cross flow or bias flow out of one porous liners
and into the other. All three (a-c) have signification effects on the wave field, either in
isolation or combined, as shown in Figures 4 to 27. Perhaps most striking is that a cross
flow with a very small Mach number M0 0:03 0:06 can significantly modify the wave field
in a mean flow with much larger Mach number M1 0:4 1:2. The reason is: (a) there is a
critical layer for acoustic-cortical waves in an unidirectional shear flow, where the local
Doppler-shifted frequency vanishes, and waves are absorbed, amplified or transformed to
another mode, as in a “valve”; (b) the cross-flow removes the critical layer, by convecting the
waves transversely across it, and creating a higher-order coupling of sound to vorticity.
For this reason, the acoustic-vortical wave equation changes from singular of second-order
in the absence of a cross-flow, to regular of the third order in the presence of bias flow.
44
International Journal of Aeroacoustics 0(0)
The effect of wall liners on sound absorption is well-known, and in the case of acousticvortical waves in a boundary layer couples to the shear flow, that can also absorb (or
amplify) waves. Thus, whereas the uniform stream conserves (or convects) the acoustic
energy, a shear flow exchanges energy with acoustic-vortical waves. In the present case,
the same free stream is matched to two boundary layers, so a thin boundary layer has large
vorticity and a thick boundary layer has weaker vorticity. Either a thick boundary layer or a
strong vorticity, combined with wall liners and bias flow can lead to a variety of waveforms,
illustrated in Figures 4 to 27 for a range of parameters of the problem.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or
publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or
publication of this article: This work was supported by the product development team at Free Field
Technologies S.A. and by FCT (Foundation for Science and Technology) through IDMEC (Institute
of Mechanical Engineering), under LAETA Pest – OE/EME/LA022.
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