-
J. Fluid Mech. (2003), vol. 485, pp. 307335. c 2003 Cambridge
University PressDOI: 10.1017/S0022112003004518 Printed in the
United Kingdom
307
The absorption of axial acoustic waves by aperforated liner with
bias ow
By JEFF D. ELDREDGE AND ANN P. DOWLINGDepartment of Engineering,
University of Cambridge, Trumpington Street,
Cambridge CB2 1PZ, UK
(Received 23 August 2002 and in revised form 29 January
2003)
The eectiveness of a cylindrical perforated liner with mean bias
ow in its absorptionof planar acoustic waves in a duct is
investigated. The liner converts acoustic energyinto ow energy
through the excitation of vorticity uctuations at the rims of
theliner apertures. A one-dimensional model that embodies this
absorption mechanismis developed. It utilizes a homogeneous liner
compliance adapted from the Rayleighconductivity of a single
aperture with mean ow. The model is evaluated by comparingwith
experimental results, with excellent agreement. We show that such a
system canabsorb a large fraction of incoming energy, and can
prevent all of the energy producedby an upstream source in certain
frequency ranges from reecting back. Moreover,the bandwidth of this
strong absorption can be increased by appropriate placementof the
liner system in the duct. An analysis of the acoustic energy ux is
performed,revealing that local dierences in uctuating stagnation
enthalpy, distributed overa nite length of duct, are responsible
for absorption, and that both liners in adouble-liner system are
absorbant. A reduction of the model equations in the limit oflong
wavelength compared to liner length reveals an important parameter
grouping,enabling the optimal design of liner systems.
1. IntroductionPerforated liners are used in engineering systems
for their ability to absorb sound. In
particular, modern combustion systems such as gas turbines and
jet engines are oftenoperated under conditions that make them
susceptible to combustion instabilities,and by including a passive
damping device, such as a liner, these acoustically
driveninstabilities can be suppressed. The high temperatures
present in the jet ows of thesesystems require that a bias ow be
blown through a perforated liner to protect the wallsof the duct
enclosing the jet. Remarkably, this bias ow has been found to be
capa-ble of providing the additional benet of improving the
eectiveness of the liner asan absorber of sound. In the present
work we investigate the eectiveness of a per-forated liner with
bias ow in absorbing planar acoustic waves propagating in a ductof
circular cross-section.
Early work considered the behaviour of perforated liners in the
absence of meanow. The reection and transmission properties of an
innite plane perforated by ahomogeneous array of orices were
examined by Ffowcs Williams (1972). Leppington& Levine (1973)
performed the same analysis with the help of an integral
equation,which allowed a correction to the Ffowcs Williams result.
They also considered thecase when the screen is backed by an innite
rigid wall. Acoustic liners are neverinnite, however, and the
acoustic properties of a semi-innite perforated screen were
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308 J. D. Eldredge and A. P. Dowling
investigated by Leppington (1977) using a WienerHopf approach.
In the absenceof ow, no damping mechanisms are present in the
linear acoustic equations; onlythe inclusion of nonlinear eects can
predict dissipation of sound. For example,Cummings (1983)
considered power losses due to high-amplitude sound
transmissionthrough orice plates, which required that nonlinear
terms be included in theanalysis.
If the holes in a perforated screen are suciently separated
relative to theirdiameter, then they nearly behave as though in
isolation. Consequently, theoreticalinvestigations of perforated
screens can utilize the signicant amount of work carriedout in
exploring the acoustic properties of a single orice. Howe (1979b)
developedan expression for the Rayleigh conductivity of an aperture
through which a high-Reynolds-number ow passes. In such a
conguration, viscous eects are limited to theseparation of the ow
from the rim of the aperture. The harmonic pressure dierenceacross
the orice thus leads to the periodic shedding of vorticity, and
consequentlyacoustic energy is converted into mechanical energy,
which is subsequently dissipatedinto heat. This mechanism is
linear, in the sense that the fraction of incidentsound energy that
is absorbed is independent of the amplitude of the sound.
Thephenomenon is also present in ow issuing from a nozzle (Howe
1979a), trailing edgeow (Howe 1986b) and abrupt area expansions
(Dupe`re & Dowling 2000). In eachexample, an unsteady Kutta
condition is applied to determine the magnitude of theuctuating
vorticity. The applicability of this condition was addressed by
Crighton(1985), who discussed its consequences in several contexts,
including the input ofacoustic energy into the vortical motion of a
shear layer.
The vortex shedding mechanism was studied further by Hughes and
Dowling,who presented analyses of screens with regular arrays of
slits (Dowling & Hughes1992) and circular perforations (Hughes
& Dowling 1990) with mean bias ows, tosuppress the screech
instability. They showed that if a rigid backing wall is
included,it is theoretically possible to absorb all impinging sound
at a particular frequency,because reection from the wall allows
substantially more interaction between thesound and screen. The
perforated liner arrangement was investigated experimentallyby Jing
& Sun (1999), with an additional term included in the model to
account forscreen thickness. Their results revealed that the
thickness of the screen is crucial, andin subsequent work (Jing
& Sun 2000) they numerically solved the governing equationfor a
single aperture in a thick plate. Their results reduce to those of
Howe (1979b) inthe limit of zero thickness. These investigations
only considered acoustic waves thatimpinge upon a perforated
screen. Many acoustically driven instabilities primarilycontain
axial acoustic modes, however, and the present investigation
focuses on theabsorptive properties of a cylindrical perforated
liner in the presence of planar ductmodes.
Cellular acoustic liners, which consist of regular arrays of
cavities, have also beeninvestigated. These liners rely primarily
on Helmholtz resonance of the cavities forabsorption. Ko (1972)
performed an analysis of a circular duct with such a liner, witha
uniform mean ow present in the duct, using a nite dierence
approximation of theequations. While most theoretical
investigations have been carried out in the frequencydomain,
Sbardella, Tester & Imregun (2001) used a time-domain approach
to developa one-dimensional model for the acoustic response of a
typical liner cavity, and coupledthis model to the solution of the
two-dimensional duct ow equations. Tam et al.(2001) used direct
numerical simulation to explore the dissipation mechanisms of
asingle liner cavity in detail, and conrmed previous observations
that vortex sheddingis the dominant mechanism of absorption for
incident waves of high amplitude. The
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The absorption of axial acoustic waves by a perforated liner
with bias ow 309
u u
B+u
Bu
r
Liner withcompliance 1
v 1
L Ld
B+d
Bd
x
B+d
BdRd =
CL
Figure 1. Schematic of acoustic waves and ow quantities in a
lined duct.
use of bias ow to improve the performance of acoustic liners has
recently beeninvestigated by Follet, Belts & Kelly (2001).
The present investigation will consist of both modelling and
experiment. Ourmodel will be based on the Rayleigh conductivity of
Howe (1979b), and like Hughes& Dowling (1990) and Jing &
Sun (1999) we will develop an eective liner compliancebased on the
principle that unsteady vortex shedding from the aperture rims is
theprimary mechanism for absorption. In 2 the model equations will
be developed, andthe treatment of dierent liner congurations will
be discussed. In 3 the nature ofliner absorption will be analysed,
which will provide useful insight for designing moreeective liners.
The model equations will be solved and compared with
experimentalresults in 4. Finally, in 5 we will identify important
parameters that enable theoptimal design of liner systems, which we
will discuss at length.
2. ModelA quasi-one-dimensional lined-duct model is developed in
this section, beginning
with the derivation of a set of equations for the acoustic
quantities in a duct with ageneral wall compliance. The compliance
has been dened by Leppington (1977) asthe ratio of the normal
derivative of a quantity on a surface to its jump across
thesurface. In the case of the stagnation enthalpy perturbation,
which we use here,
B
n= [B ]0
+
0, (2.1)
where n is the wall normal. We will construct an eective
compliance for the linerbased on the results of Howe (1979b) for a
single aperture with steady ow.
2.1. Lined-duct equations
Consider a circular duct of uniform cross-section, consisting of
two sections with rigidwalls, separated by a lined section for
which the compliance is uniformly equal to 1(see gure 1). A steady
ow of velocity uu is present in the upstream duct, as wellas a
steady inward bias ow through the liner of uniform average velocity
v1. Thesubscript on v1 denotes it, as well as other quantities and
parameters, as describingliner 1, the innermost liner. Another
liner would be denoted by subscript 2, and soon. Both mean ows are
small and the stagnation temperatures associated with themare
equal. Thus, the mean density can be uniformly approximated by and
the meanspeed of sound by c. If the length of the lined section is
L, and the cross-sectionalarea and circumference of the duct are Sp
and C1 respectively, then the downstreamow velocity is ud = uu +
(C1L/Sp)v1. As there are no circumferential variations inthe ow or
boundary conditions, the problem is axisymmetric.
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310 J. D. Eldredge and A. P. Dowling
The axial position is denoted by x, and x = 0 coincides with the
interface of thelined section and the upstream duct. Harmonic
pressure uctuations are produced inthe upstream section, of
suciently low frequency that only plane waves are allowedto
propagate. Because of the mean ow, it is more natural to describe
the acousticwaves in terms of uctuating stagnation enthalpy, B = p/
+ uu, where p is theuctuating pressure and u is the uctuating axial
velocity. Thus, in the upstreamsection, these uctuations have the
form
B (x, t) = B(x) exp[it] = B+u exp[i(t k+u x)] + Bu exp[i(t + ku
x)], x < 0,(2.2a)
where is the angular frequency and k+u and ku are the forward-
and backward-
travelling wavenumbers, respectively. We adopt the usual
convention that only thereal components of these uctuations are
physically relevant. In the downstreamsection, the uctuations are
of the same form,
B (x, t) = B(x) exp[it] = B+d exp[i(t k+d x)] + Bd exp[i(t + kd
x)], x > L.(2.2b)
The two amplitudes in the downstream section, Bd , are related
by a known boundarycondition. A generic relationship will be
assumed, Bd =RdB
+d , where Rd is the
reection coecient. In 2.3 we will discuss the specic form of Rd
.We introduce the dimensionless variables
x x/L, t ct/L, u u/c, v v/c,B B/c2, /, p p/(c2),
}(2.3)
and dene the upstream and downstream Mach numbers, Mu = uu/c and
Md = ud/c,respectively, and the bias ow Mach number, Ml,1 = v1/c.
The one-dimensionalinviscid equations of motion give the
wavenumbers as
k+u =k
1 + Mu, ku =
k
1 Mu , (2.4a, b)
k+d =k
1 + Md, kd =
k
1 Md , (2.4c, d )
where k /c. The velocity uctuations in each of these two
sections also follow,with amplitudes
u(x) =B+u
1 + Muexp(ik+u Lx) B
u
1 Mu exp(iku Lx), x < 0, (2.5a)
u(x) =B+d
1 + Mdexp(ik+d Lx) B
d
1 Md exp(ikd Lx), x > 1. (2.5b)
In the lined region 0 < x < 1, the stagnation enthalpy and
velocity have the form
B(x, t) = B + B(x) eikLt , (2.6a)
u(x, t) = u(x) + u(x) eikLt . (2.6b)
Note that the mean axial velocity varies with x. To rst
order,
u(x) = Mu +C1L
SpMl,1x. (2.7)
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The absorption of axial acoustic waves by a perforated liner
with bias ow 311
To connect the three sections, we require that the stagnation
enthalpy and velocityperturbations match at x = 0 and x = 1:
B(0) = B+u + Bu , (2.8a)
u(0) =B+u
1 + Mu B
u
1 Mu , (2.8b)
B(1) = B+d (exp(ik+d L) + Rd exp(ikd L)), (2.8c)
u(1) = B+d
(exp(ik+d L)
1 + Md Rd exp(ik
d L)
1 Md)
. (2.8d)
We will develop equations in x [0, 1] for B(x) and u(x) by
appealing to a balanceof mass and momentum with respect to a
control volume at x that spans the ductcross-section and has
innitesimal axial length, dx. First we dene a continuousow
distribution through the liner, v1(x, t), positive for inward ow.
As with otherquantities, v1 is the sum of a mean part and harmonic
perturbation,
v1(x, t) = v1 + v1(x) eikLt . (2.9)
A balance of mass across the control volume leads to
t+
x(u) =
C1L
Spv1. (2.10)
The ow through the liner acts like a source (or sink) of mass.
From momentumbalance we obtain
u
t+ u
u
x+
p
x= 0, (2.11)
which is simply the one-dimensional Euler equation.By
introducing the mean-harmonic forms of the quantities, and matching
terms with
linear dependence on uctuating components, we can form acoustic
perturbations ofequations (2.10) and (2.11). We also assume that
the uctuations are isentropic, so = B uu. We arrive at
ikLB + 2udB
dx+ (1 u2)du
dx=
C1L
Spv1, (2.12a)
ikLu +dB
dx= 0. (2.12b)
The equations are more amenable to solution if we express them
in terms of the newcharacteristic quantities,
+ 12(1 + u)[B + (1 u)u], (2.13a)
12(1 u)[B (1 + u)u]. (2.13b)
In an unlined duct, plane waves obey the relation u = B /(1M)
and the quantitiesrevert to (+, ) = (B, 0) for forward-travelling
waves or (+, ) = (0, B) forbackward-travelling waves. When these
quantities are substituted into equations(2.12a) and (2.12b), we
arrive at
d+
dx= ikL
1 + u+ +
1
2
C1L
Spv1, (2.14a)
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312 J. D. Eldredge and A. P. Dowling
d
dx=
ikL
1 u 1
2
C1L
Spv1. (2.14b)
Through equations (2.8ad ), the boundary values of + and are
+(0) = B+u , +(1) = B+d exp(ik+d L), (2.15a, b)
(0) = Bu , (1) = Bd exp(ikd L). (2.15c, d )
Since the equations are rst order, we need only enforce two
boundary conditions. Itis sucient to assume that the amplitude, B+u
, of the incident wave is known, relativeto which all other
amplitudes are measured. The downstream reection depends onlyon
geometry and mean ow conditions, and can be used to dene a linear
relationshipbetween uctuating quantities. We thus enforce
+(0) = B+u , (2.16a)
(1) exp(ikd L) +(1)Rd exp(ik+d L) = 0. (2.16b)Equations (2.14a,
b) are not closed, however, and the uctuating liner ow, v1 must
be considered. The assumed compliance of the liner, 1, implies a
local relationshipbetween v1 and the dierence in stagnation
enthalpy uctuations across the liner.Using (2.1) in conjunction
with the linearized momentum equation in the normaldirection and
the relation B = + + , we arrive at
v1(x) =1
ikL[B1(x) +(x) (x)], (2.17)
where B1 is the uctuating part of the stagnation enthalpy
external to the liner,B1(x, t) = B1 + B1(x) exp(ikLt). The
behaviour of B1 depends upon the specicconguration of the duct
system. We consider three:
Conguration 1. Open exterior. When the liner is exposed to the
ambientenvironment it is satisfactory to assume that the pressure
uctuations (and hencestagnation enthalpy uctuations, since the two
quantities are proportionally relatedin the absence of mean ow) on
the outside of the liner are zero. Thus, in this con-guration,
v1(x) = 1ikL
B(x) = 1ikL
[+(x) + (x)], (2.18)
and the system (2.14a, b) and (2.18) is closed. Note that for
this relation to beappropriate, the exterior of the liner need only
be exposed to a region in whichpressure uctuations are negligibly
small. Thus, a large enclosing cavity of volumeV C1L/k is ensured
to behave suciently similarly to a plenum to justify
theassumption.
Conguration 2. Annular cavity enclosed by rigid wall. In this
conguration the lineris surrounded by an annular cavity formed by a
larger rigid cylinder. The cavity isof the same axial length as the
liner, with rigid walls at each end. If the cavity issuciently
shallow, then the acoustic equations in this region are
one-dimensional,as in the duct. Using a similar balance of mass and
momentum through an annularcontrol volume of length dx leads to the
non-dimensional equations
dB1dx
= ikLu1, (2.19a)du1dx
= ikLB1 C1LSc
v1, (2.19b)
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The absorption of axial acoustic waves by a perforated liner
with bias ow 313
where Sc is the cross-sectional area of the annular cavity.
Together with the rigid-wallboundary conditions
u1(0) = 0, u1(1) = 0, (2.20)
equations (2.14a, b), (2.17) and (2.19a, b) now form a closed
system of four dieren-tial equations for the four unknowns +, , B1
and u1. Note that such a cong-uration is not practically feasible
if a bias ow is present, as there must be an inletin the cavity for
the mass ow. However, if this inlet is choked, then the
rigid-wallapproximation is likely to be sucient.
Conguration 3. Annular cavity enclosed by second liner. This
conguration is verysimilar to the previous one, except that the
annular cavity is now formed by a secondperforated liner. This
arrangement might be found in a cooling ow system, in whichthe
outer skin supplies jets that impinge upon the outer surface of the
inner liner.We will assign compliance 2 and circumference C2 to the
outer cylinder. Then theequations in the cavity are
dB1dx
= ikLu1, (2.21a)du1dx
= ikLB1 C1LSc
v1 +C2L
Scv2, (2.21b)
where v2 is the inward uctuating ow through the outer liner,
related to the stagnationenthalpy dierence across the liner by
v2(x) =2
ikL[B2(x) B1(x)]. (2.22)
The boundary conditions for the cavity quantities are the same
as before, (2.20). Thesystem of equations (2.14a, b), (2.17) and
(2.20)(2.22) is closed with some treatmentof B2, the uctuating
stagnation enthalpy external to the second liner, involving oneof
the three congurations discussed. Note that, through conservation
of mass, themean bias ows through each liner are related,
Mh,1 =C2
C1
2
1Mh,2. (2.23)
It is important to note that this one-dimensional model is only
valid when theplanar mode is the only propagating mode in the duct.
For a circular duct, this limitsthe maximum frequency to
kL < 1.841
(1
2
C1L
Sp
), (2.24)
to avoid the cut-on of the lowest azimuthal mode. Regardless of
the conguration ofthe ductliner system, the system of equations
constitutes a boundary value problem.In principle, an analytical
solution for this linear system is possible, though with alinearly
increasing mean duct ow it requires some eort. Even if the
variability of themean duct ow is neglected, the solution is too
complicated to be revealing (thoughwe do present the solution for a
liner in Conguration 1 with no mean duct ow inthe Appendix). Thus,
we solve the system numerically, using a shooting method toenforce
boundary conditions at either end of the lined section (see, e.g.,
Press et al.1986). Before solution is possible, however, an
expression for the liner compliancemust be developed.
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314 J. D. Eldredge and A. P. Dowling
2.2. Liner compliance
The acoustic behaviour of a single aperture can be used to
develop an expression forthe compliance of a homogeneous screen of
such apertures, provided that the distance,d , between neighbouring
apertures is large compared with their diameters, 2a, andthe
acoustic wavelength, , is much larger than the inter-aperture
distance. Thus, werequire ka kd 1. This approach has been used by
Hughes & Dowling (1990),who adapted the Rayleigh conductivity
of a single aperture in a plane proposed byHowe (1979b) to
construct a smooth compliance for a perforated screen with biasow.
We will use the same approach in the present model.
The Rayleigh conductivity relates the uctuating aperture volume
ux, Q =Q exp(ikLt), to the uctuating dierence in stagnation
enthalpy across the aperture.In dimensionless form,
K = ikL Q[B]0
+
0. (2.25)
For simplicity, we assume that the screen is composed of a
square grid of apertures,with equal distance d between adjacent
apertures. The number of apertures per unitarea in the screen is N
= 1/d2 and the uctuating liner velocity is equal to v =
NQ.Comparing (2.25) with (2.1), with consideration of the
linearized momentum equation,the compliance has a simple
relationship with the conductivity of a single
componentaperture,
= NL2K = KL2/d2. (2.26)
The expression for the conductivity of a single aperture in an
innitely thin wall,derived by Howe (1979b), is here adapted to the
convention of positive exponent inthe harmonic factor
exp(ikLt),
Ka =2a
L( + i), (2.27)
where a is the radius of the aperture, and and are given by
equations (3.14)of Howe (1979b). Each is dependent on only the
Strouhal number of the ow,St = a/Uc, where Uc is the convection
velocity of vortices shed from the aperture rim.Howe argues that
this velocity is approximately equal to the mean velocity in
theaperture. If is the open-area ratio of the perforated screen,
a2/d2, thenthe aperture Mach number, Mh Uc/c, is related to the
mean liner ow throughMh = Ml/ , and
St kaMh
=ka
Ml. (2.28)
Howes model is based on a ow that behaves ideally, except
insofar as viscosity isresponsible for the creation of vorticity at
the sharp rim of the aperture. It neglectsmany real ow features,
such as wall thickness and accompanying boundary layerdissipation.
The inertial eect of liner thickness can be approximately
accountedfor by regarding the stagnation enthalpy dierence across
the liner as a sum ofthe contributions to unsteady vortex shedding
and acceleration of the uid in theaperture. This leads to the
following expression for the total compliance:
1
tot=
a2
L21
Ka+
t
L, (2.29)
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The absorption of axial acoustic waves by a perforated liner
with bias ow 315
where t is the liner thickness, which may include a correction
for end eects. Jing& Sun (1999) have used this approach in
their model, and their comparison withexperimental results is
favourable. However, their more recent work (Jing & Sun2000)
has demonstrated that the eective thickness decreases with
increasing biasow, behaviour which is absent from (2.29). Bellucci,
Paschereit & Flohr (2002) havedeveloped a model for the
acoustic impedance of a perforated screen backed by arigid wall
that incorporates several eects, including viscous losses in the
aperture andan allowance for reactance of the external ow. Their
model agrees well with theirexperimental results, though they
slightly overpredict the magnitude of reection in athick screen at
resonance. Because our goal is to predict the absorptive properties
ofa perforated liner, embodied in the imaginary part of the
compliance, the thicknessis less important than to model accurately
and expression (2.29) is used. It will bedemonstrated that this
approach is adequate.
It is worth noting that the Rayleigh conductivity of Howe
(1979b) is strictly validin the presence of a mean bias ow only,
and not with an additional grazing ow.In subsequent work, Howe,
Scott & Sipcic (1996) have numerically computed theRayleigh
conductivity of an aperture with mean grazing ow only. We
hypothesizethat the bias ow creates a thin boundary layer adjacent
to the liner, in which the biasow is dominant. Vorticity uctuations
at the aperture rim are assumed to convectnormally from the liner,
and cannot interact with the downstream aperture rim andgenerate
sound as is possible in the analysis of Howe et al. (1996). The
convectedvorticity may eventually reach the edge of the boundary
layer and interact with themean duct ow, but will probably lose
coherence in the process and have negligibleeect on apertures
downstream.
2.3. Downstream reection coecient
In boundary condition (2.16b), the reection coecient, Rd ,
appears and we nowdiscuss its form. In the trivial case of a
semi-innite downstream section, no acousticwaves are reected and Rd
=0. For an unanged open end, which is more likely tobe encountered
in practice, previous investigations can be relied upon to develop
anappropriate expression. In the absence of a mean duct ow, the
reection is nearlyperfect save the small energy lost to radiation.
Levine & Schwinger (1948) havesolved this problem using a
WienerHopf approach. For low frequencies, the specicacoustic
impedance of the open end is well approximated by
(kRp) =14(kRp)
2 + i0.6kRp, (2.30)
where Rp is the radius of the duct. The reection coecient is
related to by
Rd = (kRp) 1 (kRp) + 1
exp[i2k(L + Ld)], (2.31)
where Ld/L is the scaled downstream duct length. Thus, Rd can be
approximated tosecond order in kRp by
Rd = (1 12 (kRp)2) exp[i2k(L + Ld + )], (2.32)where 0.6Rp is the
end correction.
When mean ow is present, acoustic energy will be removed from
the duct throughradiation as well as through excitation of jet
instability waves (Cargill 1982a, b; Munt1977, 1990). The latter
eect can be accounted for by imposing a Kutta conditionat the edge.
Cargill (1982a) has solved the problem using the WienerHopf
technique
-
316 J. D. Eldredge and A. P. Dowling
and assumed kRp 1 to arrive at the following formula for Rd ,
second-order-accuratein kRp:
Rd = (1 M2)g (1 M)(1 M2)g + (1 + M)
(1 12 (kRp)2) exp[i2k((L+Ld)/(1 M2)+ )].(2.33)
The function g = (g1 + ig2)/M is related to the jet instability
and is given by Cargill(1982a); g1 and g2 depend only on the duct
Strouhal number, StRp = kRp/M . For lowM and large StRp , Rienstra
(1983) found that the end correction is approximately thesame as in
the no-ow case. Note that equation (2.33) has been adjusted to
representthe reection of stagnation enthalpy, rather than pressure.
For low M , it can beapproximated by
Rd = (1 2g1
g21 + g22
M
)(1 1
2(kRp)
2)exp[i2k(L + Ld + )]. (2.34)
The magnitude of this formula is in excellent agreement with the
results obtained byMunt (1977) numerically.
Peters et al. (1993) have demonstrated that (2.34) agrees well
with experimentalresults for StRp < 1. For StRp 2 (as in most
cases considered experimentally in thepresent paper), the coecient
of M in (2.34) tends to 1.1, though the experimentalresults are
less supportive of the formula in this range of Strouhal number.
For allcases under our consideration, however, the downstream duct
Mach number will bequite small, Md 1, and therefore the precise
dependence of the reection coecienton the ow is not crucial. Under
such a condition, the absorption results fromequations (2.32) and
(2.34) are indistinguishable.
3. Acoustic energy considerationsTo learn more about the nature
of the liner absorption, it is useful to analyse the
ux of energy in the lined section. The acoustic energy ux vector
in a moving owis given by (Morfey 1971)
Ii = B (ui) = 14[B (ui) + B (ui)], (3.1)where the angle brackets
denote a time average of the product of the real componentsof these
quantities, the superscript signies the complex conjugate, and the
subscripti denotes the direction (i = 1 corresponds to the axial
direction, x). We now derivean expression for the rate of change of
I1 with x. First, note that our denition ofthe quantities + and
provides for a simple relation with the term in brackets
in(3.1),
I1 =12(|+|2 ||2). (3.2)
This relation attaches physical signicance to + and . They
represent the forward-and backward-transmitted sound intensities in
a duct with general wall impedance.The derivative of this relation,
when combined with equations (2.14a, b), leadsto
dI1dx
=1
4
C1L
Sp(+v1 +
+v1 + v1 + v1). (3.3)
The rst two terms in this expression represent the rate of
change of the forward-travelling energy, and the latter two terms
correspond to the the backward-travelling
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 317
energy change. The liner ow v1, through the compliance relation
(2.17), couples theserates of change, ensuring that
forward-travelling energy is continuously partitionedinto a
transmitted portion, a portion that is absorbed by the liner and a
portion thatis reected upstream, and similarly for the
backward-travelling energy.
The quantities + and are replaced by their denitions (2.13a, b)
to expressrelation (3.3) in terms of the primitive quantities. We
arrive at
dI1dx
=C1L
SpB v1. (3.4)
When we integrate this expression over the length of the liner
and multiply by theduct area Sp , we arrive at the expected energy
balance,
Sp[I1(1) I1(0)] = C1L 10
B (x)v1(x) dx, (3.5)which accounts for the net change in duct
energy by the ux of energy through theliner. Because v1 has been
dened as positive for inward ow, when the right-handside of
equation (3.5) is negative, then energy has been absorbed by the
liner. Usingrelation (3.2) with the boundary values of + and ,
then
12Sp(|B+d |2 + |Bu |2 |B+u |2 |Bd |2) = C1L
10
B (x)v1(x) dx. (3.6)The acoustic absorption of the liner, , is
dened as the net energy absorbed by theliner, scaled by the energy
incident upon the lined section. This latter energy is
I1,in 12Sp(|B+u |2 + |Bd |2), (3.7)and thus
1 |B+d |2 + |Bu |2
|B+u |2 + |Bd |2 = 2C1L
Sp
10
B (x)v1(x) dx|B+u |2 + |Bd |2 . (3.8)
We now look more closely at the right-hand side of (3.5) to
learn more about thenature of the liner absorption. We will assume
a conguration of two perforatedliners, with compliances 1 and 2,
respectively, the second liner being exposed to theambient
environment. If we use the compliance relation (2.17), then this
leads to 1
0
B (x)v1(x) dx = 10
B 1(x)v1(x) dx + kL2 Im(
1
1
) 10
|v1(x)|2 dx. (3.9)An analysis in the annular cavity, similar to
the analysis in the duct that led to (3.5),produces on application
of the boundary conditions u1(0) = u1(1) = 0, 1
0
B 1(x)v1(x) dx = C2C1 10
B 1(x)v2(x) dx. (3.10)
Finally, we use the second compliance relation (2.22) with B2 =
0, to arrive at 10
B (x)v1(x) dx = kL2 Im(
1
1
) 10
|v1(x)|2 dx + kL2
C2
C1Im
(1
2
) 10
|v2(x)|2 dx.(3.11)
It is apparent from equation (3.11) that, provided that the
imaginary part of the com-pliance of each liner is negative, then
there will be a net outward ux of energy through
-
318 J. D. Eldredge and A. P. Dowling
Loudspeakers
Transducers Transducers
Air inlet
Cavity
Dataacquisition
Innerliner
Outerliner
Plenum
Signalgeneratorand amp.
Figure 2. Conguration of the perforated liner experiment.
the liner, and acoustic energy in the duct will be absorbed.
From equation (2.29) wehave
Im
(1
)=
2
a
L
2 + 2. (3.12)
Because is positive for all Strouhal numbers when the Rayleigh
conductivity ofHowe (1979b) is used, the expression is always
negative and thus energy is alwaysabsorbed. The amount of energy
absorbed will depend upon the magnitudes of bothterms in (3.11),
prediction of which is left to the numerical solution of the
modelequations.
It is worth contrasting the present model with that of Hughes
& Dowling (1990).In their work, an expression for the reection
coecient of impinging acoustic wavesis developed. The absorption
mechanism is the same as that considered here, andthe same Rayleigh
conductivity is used to develop an eective screen
compliance.However, their reection is also dependent upon a
so-called resonance parameter,Q (kd cos )2l/2a, where l is the
distance between the screen and backing wall and is the angle of
incidence of acoustic waves (i.e. for normally incident sound, =
0).The imaginary volume behind each aperture behaves like an
individual Helmholtzresonator, and Q measures the proximity of a
liner to resonance of this volume. Bythis theory, Q = 0 for waves
that travel parallel to the liner, and consequently noabsorption is
predicted. The crucial dierence between this and the present
theoryis our use of relation (2.17) as a local relationship in x.
Rather than depending ona Helmholtz-type resonance for maximum
absorption, the integrated eect of localuctuating stagnation
enthalpy dierence is responsible.
4. Experiment4.1. Experimental set-up
The experimental apparatus is shown in gure 2. Two equal-length
sections of rigid-walled pipe of diameter 12.7 cm and length 80 cm
were separated by a section ofequal diameter and length 17.8 cm,
perforated by a regular lattice of holes of diameter0.75 mm and
spacing 3.3 mm (i.e. 1 = 0.040). This perforated section was
surroundedby a duct of diameter 15.2 cm and equal length,
perforated by a dierent array ofholes of diameter 2.7 mm and
spacing 17.0 mm (2 = 0.020). The thickness of bothperforated liners
was 3mm. The lined section was encased in a large cavity, into
which
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 319
was fed a steady supply of air by a centrifugal pump to produce
bias ow throughthe liner (measured by a Pitot probe in the cavity
inlet), up to a maximum rate of0.04 m3 s1. One end of the duct
system had four loudspeakers uniformly arrangedabout the pipe
circumference; the other emptied into a large plenum that allowed
airto be drawn through the system independently of the air fed
through the lined section.The rate at which this air was drawn
(measured by a Pitot probe in the upstreamduct section) was limited
by ow noise to 0.45 m3 s1. The centrelines of the holesin the inner
liner were tilted downstream by approximately 45 so that ow
passingthrough them would travel downstream in the duct, even in
the absence of a meanduct ow; for this reason the eective thickness
of this liner was larger by a factor of2.Harmonic acoustic waves of
magnitude between 90 and 120 dB, and frequency
between 100 and 700 Hz, were produced in the duct. This range is
well below thecut-on frequency of the rst radial mode at 3290 Hz.
Three Kulite transducers werearranged on both the upstream and
downstream sections to record the pressureuctuations. From these
measurements the amplitudes of the left- and
right-travellingacoustic waves in each section were calculated
using a two-microphone technique(Seybert & Ross 1977), and thus
the fraction of energy entering the lined section thatwas absorbed
could be deduced. The two-microphone formula for the amplitudes
P+
and P of these travelling waves in a given section is
P+ =p1 exp[ikx1/(1 M)] p2 exp[ikx2/(1 M)]exp[i2kx1/(1 M2)]
exp[i2kx2/(1 M2)] , (4.1a)
P =p1 exp[ikx1/(1 + M)] p2 exp[ikx2/(1 + M)]exp[i2kx1/(1 M2)]
exp[i2kx2/(1 M2)] , (4.1b)
where p1 and p2 are the measured pressures at transducer
positions x1 and x2,respectively. With three transducers in each
section, three separate pairs could beused to obtain the same
amplitudes and thus improve the accuracy. After calculatingthese
amplitudes from both the upstream and downstream sections, equation
(3.8)was used to compute the absorption, using the relationship
between the pressureand stagnation enthalpy in axial plane waves, B
= p(1 M), with the positive signchosen for waves travelling in the
mean ow direction and the negative sign for thosetravelling against
it.
The frequency and ow conditions were held constant for each
experiment. Thetransducer signals were low-pass ltered, with the
lter cut-o set at 1 kHz. Thesignals were sampled at a rate of 5
kHz. For the data analysis, 30 contiguous timesegments of 1024
samples were recorded. All transducers were calibrated for gainand
phase relative to a reference channel, and the reference channel
was calibratedabsolutely using a pistonphone. The signal coherence
between reference and responsechannels at the frequency of interest
was always larger than 0.95. Though the ow-induced noise in the
apparatus was signicant in some cases, the sound pressure
levelproduced by the loudspeakers was suciently high to ensure a
strong peak signal forall transducers not situated at pressure
nodes.
The three transducers in each section were separated by 20 cm
for all experimentsbelow 600 Hz, and by 10 cm for 600 Hz and above.
Seybert & Soenarko (1981) haveshown that the two-microphone
formula is prone to error as the separation of twotransducers
approaches half a wavelength. Thus, no pair of transducers was used
inthe formula that was separated by more than 70% of half a
wavelength. Furthermore,
-
320 J. D. Eldredge and A. P. Dowling
the outer transducers were always at least 20 cm from the ends
of the duct, by whichdistance the rst higher-order mode would have
decayed by 100 dB at 1000 Hz.
For a given set of experiments at several frequencies or ow
conditions, the outputof the loudspeakers was adjusted so that the
sound pressure level at the midpoint ofthe upstream duct was
constant.
4.2. Results
For comparison with experimental results, the model equations
were solved for eachset of ow conditions over the frequency range
0700 Hz. Conguration 3 was used toclose the equations, with the
uctuations external to the outer liner assumed negligible,B 2 0.
The latter assumption is justied because the enclosure that encased
the linersystem was suciently large to provide very little stiness
to the uctuating ow inthe outer liner, particularly with the
unchoked air inlet, which allowed communicationwith the
environment. For convenience, we summarize the equations here:
d+
dx=
[ikL
1 + u(x)+
1
2
C1L
Sp
1
ikL
]+ 1
2
C1L
Sp
1
ikL( B1), (4.2a)
d
dx=
[ikL
1 + u(x)+
1
2
C1L
Sp
1
ikL
] +
1
2
C1L
Sp
1
ikL(+ B1), (4.2b)
dB1dx
= ikLu1, (4.2c)du1dx
= [ikL +
C1L
Sc
1
ikL+
C2L
Sc
2
ikL
]B1 +
C1L
Sc
1
ikL(+ + ), (4.2d)
+(0) = 1, (1) exp(ikd L) +(1)Rd exp(ik+d L) = 0, (4.2e, f )u1(0)
= 0, u1(1) = 0, (4.2g, h)
where u(x) = Mu + (C1L/Sp)Ml,1x and we have set the amplitude of
the incidentwave, B+u , equal to unity. The compliance for each
liner is given by (2.29), and thedownstream reection coecient by
(2.34). These equations are solved for x [0, 1]using the shooting
method. Once the solution is obtained within the desired
tolerance,the absorption is calculated using equation (3.8), which
can also be written in termsof + and as
=1 + (|Rd |2 1)|+(1)|2 |(0)|2
1 + |Rd |2|+(1)|2 . (4.3)Before reporting absorption results, we
check that our assumed downstream
boundary condition compares favourably with the experimental
results. Figure 3compares with theory the real and imaginary parts
of the reection coecient observedfrom experiments carried out with
a liner bias ow of Mh,1 = 0.023 and no meanduct ow. The agreement
is good, with only slight discrepancy at extrema and at700 Hz.
Figure 4 demonstrates the results with a small mean duct ow, Mu =
0.046,and liner ow, Mh,1 = 0.030. Once again, the agreement is very
good, with a similardiscrepancy at 700 Hz.
In the rst set of absorption experiments in gure 5, the mean
liner bias ow is set atMh,1 = 0.023 with no mean duct ow. The
results from experiments at sound pressurelevels of 100 dB and 115
dB are compared with results from solution of the modelequations.
The absorption undergoes large oscillations between peaks and
troughsacross the entire range of frequencies, reaching 82% at the
peaks, but dropping to
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 321
(a)2.0
1.0
0
1.0
2.00 200 400 600
Frequency (Hz)
Re(
Rd)
(b)2.0
1.0
0
1.0
2.00 200 400 600
Frequency (Hz)
Im(R
d)
Figure 3. (a) Real and (b) imaginary parts of the downstream
reection coecient forMh,1 = 0.023 and Mu = 0. Experiment: ; model:
.
(a)2.0
1.0
0
1.0
2.00 200 400 600
Frequency (Hz)
Re(
Rd)
(b)2.0
1.0
0
1.0
2.00 200 400 600
Frequency (Hz)
Im(R
d)
Figure 4. (a) Real and (b) imaginary parts of downstream
reection coecient forMh,1 = 0.03 and Mu = 0.046. Experiment: ;
model: .
as little as 10% at the minima. The similarity of the
experimental results at the twodierent sound levels conrms that
linear absorption mechanisms are dominant. Themodel predicts the
absorption very well, though there is some disagreement abovekL
1.75. The measurements at higher frequencies are more prone to
experimentalerror; the discrepancy at the largest value of kL
(corresponding to 700 Hz) is probablydue to such error, as the
relative transducer calibration exhibited a large phase shiftnear
this frequency. The disagreement at the trough of kL 1.8 can be
reasonablyattributed to high sensitivity to the degree of
interference between the left- and right-travelling waves in the
liner. At this frequency, as will be explained below, a
pressureminimum occurs at some position in the lined section. The
total absorption acrossthe liner is crucially dependent on the
magnitude of the pressure near this minimum.If this magnitude is
over-predicted by the model, then the absorption trough will
beshallower, as is seen in gure 5.
The results from a larger mean bias ow, Mh,1 = 0.041, no mean
duct ow, andsound pressure level at 100 dB are depicted in gure 6.
The same oscillatory behaviouris evident, but the peak absorption
values are only around 70% now; the minimumvalues are comparable to
the previous case, however. At lower frequencies theagreement of
the model with measured values is excellent, but the comparison
isless favourable above kL 1.7. The experimentally observed
absorption is generallylower in the upper range of frequencies, and
only rises to 55% at the nal peak,instead of the 60% predicted by
the model.
-
322 J. D. Eldredge and A. P. Dowling
0.6
0.4
0.2
0
0 0.5 1.0 1.5kL
Abs
orpt
ion
2.0
0.8
1.0
Figure 5. Absorption with varying frequency for Mh,1 = 0.023 and
Mu = 0.Experiment, 100 dB: ; experiment, 115 dB: ; model: .
0.6
0.4
0.2
0
0 0.5 1.0 1.5kL
Abs
orpt
ion
2.0
0.8
1.0
Figure 6. Absorption with varying frequency for Mh,1 = 0.041 and
Mu = 0.Experiment, 100 dB: ; model: .
We now assess the eect of a small mean duct ow on the
absorption. In gure 7 aredepicted the results of experiments
conducted with a mean duct ow of Mu = 0.046and liner bias ow of
Mh,1 = 0.030, and at two dierent sound pressure levels, 100 dBand
115 dB. The character of the absorption is very similar to that of
the previouscases, with peaks at nearly 80%. As in the rst case, no
signicant variations areobserved in the results from the two
dierent sound pressure levels, so nonlinearabsorption mechanisms
are again negligibly small. The model once again predicts
theabsorption very well.
In the nal case of this type we consider a slightly larger mean
duct ow, Mu =0.057, and a smaller mean bias ow, Mh,1 = 0.009, with
a sound pressure level of100 dB. Figure 8 depicts the results.
Though the experimental results exhibit behaviour
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 323
0.6
0.4
0.2
0
0 0.5 1.0 1.5kL
Abs
orpt
ion
2.0
0.8
1.0
Figure 7. Absorption with varying frequency for Mh,1 = 0.030 and
Mu = 0.046.Experiment, 100 dB: ; experiment, 115 dB: ; model: .
0.6
0.4
0.2
0
0 0.5 1.0 1.5kL
Abs
orpt
ion
2.0
0.8
1.0
Figure 8. Absorption with varying frequency for Mh,1 = 0.009 and
Mu = 0.057.Experiment, 100 dB: ; model: .
similar to the previous cases, the model predicts absorption
peaks that are morerounded than in previous cases. Consequently,
the agreement is poor near thesepeaks, though it is satisfactory at
the absorption minima.
The four sets of experiments described were all conducted with
constant owconditions and varying frequency. They demonstrated that
the model performs verywell in predicting the absorption over a
range of frequencies. Now we present theresults when the frequency
is held xed and the mean liner bias ow varied. No meanduct ow is
present and the sound pressure level is constant at 100 dB. The
absorptionat four dierent frequencies is depicted in gure 9.
Frequencies of kL = 0.89, 1.54 and2.11 correspond to absorption
peaks in the previous cases, and kL = 1.22 correspondsto a minimum.
In the absorption peak cases, the absorption reaches a maximum
at
-
324 J. D. Eldredge and A. P. Dowling
kL = 0.891.0
0.8
0.6
0.4
0.2
0
0 0.02 0.04 0.06
Abs
orpt
ion
kL = 1.221.0
0.8
0.6
0.4
0.2
0
0 0.02 0.04 0.06
kL = 1.541.0
0.8
0.6
0.4
0.2
0
0 0.02 0.04 0.06
Abs
orpt
ion
kL = 2.111.0
0.8
0.6
0.4
0.2
0
0 0.02 0.04 0.06
Mh, 1 Mh1
Figure 9. Absorption with varying mean bias ow and Mu = 0.
Experiment, 100 dB: ;model: .
around 82% at approximately the same ow, Mh,1 0.015, and decays
gradually asthe ow is increased. The model predicts this behaviour
well at low frequencies, butperforms poorly at the highest
frequency, kL = 2.11. At kL = 1.22, both experimentand model agree
at a nearly constant value of around 20%.
5. Liner system design5.1. Maximizing absorption
From a design standpoint, it is desirable to nd the geometric
and ow parameters formaximum absorption. Such an optimization
problem is complicated by the number ofparameters involved. For
example, for a single liner in Conguration 1, the
geometricparameters that appear in the model equations are
(dropping the 1 subscript forbrevity): the open-area ratio, ; the
scaled aperture radius, a/L; the scaled linerthickness, t/L; and
the duct geometric parameter, CL/Sp . The ow parameters are:the
scaled frequency, kL; the aperture Mach number, Mh; and the duct
Mach number,Mu. Furthermore, we will show that the absorption is
intimately connected to (andpotentially enhanced by) the downstream
conguration, so the design process shouldincorporate Rd . An
additional liner increases the complexity still further. We
willapproach the design process systematically, beginning by
identifying our goals andthen reducing the problem to its barest
essentials in a single, innitesimally thin liner.Only then will we
include a second liner in the design, developing an expression
thatdescribes the sharing of absorption between the two liners.
The absorption expression (4.3) provides a useful starting
point. Provided themagnitude of the downstream reection is less
than or equal to unity, as is normally
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 325
0.6
0.4
0.2
0
0 0.5 1.0 1.5kL
Abs
orpt
ion
2.0
0.8
1.0
Figure 10. Comparison of absorption in a duct with Mh,1 = 0.023
and Mu = 0, and withdownstream length Ld/L = 4.5: ; Ld/L = : .
true, this expression conrms what is intuitively obvious, that
absorption is large whenthe acoustic energy transmitted, |+(1)|2,
and reected, |(0)|2, by the liner systemare both small. Because the
downstream reection may not be as simple as an openend, one
approach would be to assume nothing about the downstream
congurationand optimize the liner system in a duct that extends to
innity downstream (i.e.Rd =0). This approach will lead to adequate
results, but does not exploit the potentialenhancement that arises
from most practical downstream arrangements. For example,gure 10
depicts the absorption of a liner system in both semi-innite and
nite ductcongurations. The absorption is substantially increased in
certain frequency bands.It is important to incorporate such
enhancement into the design process directly.Consequently, we will
assume that the downstream conguration allows most of theacoustic
energy to reect back towards the liner (i.e. |Rd | 1), and optimize
the linerarrangement such that the resulting absorption peaks are
as large as possible.
Because small transmission and reection coincide with large
absorption, it isnatural to enquire whether the minima of these two
quantities naturally coincide, andif not, whether we can force them
to. In the Appendix, the single-liner equations aresolved in the
absence of mean duct ow, and the solution related to the
semi-innitedownstream duct result. Relations (A 7a, b) for the
transmitted and reected energiesimmediately exhibit some essential
features of the nite-duct results. In particular, thetransmitted
wave energy, equation (A 7a), can be enhanced or diminished
dependingonly on the phase of the product Rd
(0). The smallest transmission will occur whenthis phase is an
odd multiple of ,
arg(Rd(0)) = (2n 1). (5.1)
Furthermore, larger magnitude of (0) will lead to greater
deviation of thistransmitted wave from its semi-innite result.
Insight into the reected wave amplitude is not as easy to
obtain: relation(A 7b) demonstrates that the reected energy is
dependent upon a strong couplingbetween the left- and right-going
waves in the liner. The phases of Rd
(0)
-
326 J. D. Eldredge and A. P. Dowling
and Rd (0)[((0))2 (+(1))2] appear in the denominator and
numerator,
respectively. With some rearrangement to isolate Rd , though,
the phases can beregrouped into the phases of Rd
(0) and 1 (+(1)/(0))2. Thus, minimumreection is not simply
determined by equation (5.1), but by an intractable
expressioninvolving both phases. In general, these phases are
uncorrelated, and therefore thereection and transmission minima do
not necessarily coincide. Furthermore, becauseof the complicated
dependence of the semi-innite solution on frequency, it
provesimpractical to tailor the liner so that they do coincide.
However, we shall showthat the minima do approximately coincide at
low frequency, gradually diverging asfrequency increases.
In fact, relying on the full frequency-dependent solution for
optimization is imprac-tical in many respects, so we instead follow
a simpler approach. The long-wavelengthlimit kL 0 of the
semi-innite duct solution, though not interesting in itself,
providesat least a reasonable prediction of the behaviour at low
frequency, particularly becausethis solution tends to vary more
slowly with frequency than Rd . We will thereforesubstitute the
limiting values of +(1) and (0) into the nite-duct relations (A 7a,
b),and optimize the resulting approximate absorption.
With given by equation (2.29) and t/L = 0, the liner compliance
factor /(ikL)remains non-zero as kL approaches zero:
limkL0
ikL=
1
2
Mh. (5.2)
This limit can be interpreted in terms of a loss of dynamic
head. A steady jet emergesfrom each aperture, in which the steady
perturbation of the pressure is related tothe velocity perturbation
by pj = Mjvj through a non-dimensional version ofBernoullis
theorem. If this jet loses all of its dynamic head as it empties
into the ductand dissipates, then the stagnation pressure (or
enthalpy) in the duct will be equal to itsstatic value in the jet,
B pj . The velocity perturbation of the jet is related to theliner
velocity through vj = v/ . Howe (1979b) argues that the mean jet
velocity istwice the mean aperture velocity because of the vena
contracta. Thus, we nd thatv/B /(2Mh), which is simply the negative
of /(ikL).
The liner equations in the limit of long-wavelength reduce
to
d+
dx= b(+ + ), (5.3a)
d
dx= b(+ + ), (5.3b)
where
b CL4Sp
Mh. (5.4)
A steady perturbation to the stagnation enthalpy maintains a
steady and uniformdierence across the liner, which therefore draws
a liner ow that causes the velocityperturbation to vary linearly
along the length of the liner. The semi-innite ducttransmission and
reection are thus, respectively,
+(1) =1
1 + b, (0) = b1 + b . (5.5)
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 327
Substituting these into the nite-duct expressions, and assuming
that |Rd | = 1, then
|+(1)|2 = 1(1 + b)2 + b2 2b(1 + b) cosR , (5.6a)
|(0)|2 = (1 b)2 + b2 + 2b(1 b) cosR
(1 + b)2 + b2 2b(1 + b) cosR , (5.6b)
where R arg(Rd) = arg(Rd) + 2kL is a long-wavelength
approximation ofarg(Rd
(0)). The limiting phase of 1 (+(1)/(0))2 is either (when b <
1) or0 (when b > 1), though both ensure that the minima of
reection and transmissioncoincide, which, as equation (5.1)
predicts, will occur when R = (2n 1), where n issome integer. At
these minima the transmitted and reected energies are determinedby
b. At b = 0, the liner is eectively rigid and thus both energies
are equal to theincident energy (since the reection is perfect).
Increasing b causes both thetransmitted and reected energies to
decrease steadily until b = 1/2, at whichthe reected energy is zero
but the transmitted energy is still nite. Beyondb = 1/2, the
transmitted energy continues to decay, approaching zero as b ,but
the amount of reected energy increases. Note that, because the
transmittedenergy never reaches zero, the liner system can never
absorb all incident acousticenergy.
Substituting (5.6a, b) into the absorption expression (4.3), we
arrive at
=2b(1 cosR)
1 + b(b + 1)(1 cosR) . (5.7)
As expected, the absorption maxima occur at the coincident
minima of reection andtransmission, at R = (2n 1). The value of b
that gives the greatest absorptionat these maxima is 1/
2, at which = 0.8284. Therefore, for a single liner
conguration, the liner parameters should as closely as possible
obey(Mh
)opt
=CL/Sp
22
. (5.8)
This optimum relation will become less valid at higher
frequency, particularly if Rddoes not vary signicantly faster with
kL than the semi-innite duct results (forinstance, when Ld/L 1).
But it should at least provide a reasonable guideline forchoosing
parameters. Note that the ratio of hole diameter to liner length,
a/L, doesnot appear in this expression.
When there are two liners, the absorption load is shared by both
liners. Again,we make the assumption that the semi-innite
downstream duct solution varies moreslowly with frequency than Rd .
Because of the relationships between the geometricand ow
parameters, it can be veried that the dierential equation for the
acousticvelocity between the liners (equation (2.21b)) reduces
to
du1dx
= 2b (C1/C2)2
1 (C1/C2)2{[
1 +
(C2
C1
)2(2
1
)2]B1 +
}, (5.9)
where b is dened as before. The stagnation enthalpy uctuations
both in the ductand between the liners are uniform, and thus the
solution of (5.9) must vary linearly.However, to satisfy the
no-throughow conditions at both ends of the liner, therecan be no
variation and the term in curly brackets in (5.9) must vanish. Thus
the
-
328 J. D. Eldredge and A. P. Dowling
stagnation enthalpy uctuations on either side of the inner liner
are related by[1 +
(C2
C1
)2(2
1
)2 ]B1 =
+ + = B. (5.10)
Substituting this into the corresponding duct equations, we
arrive at
d+
dx= b
(1/2)2(C1/C2)2 + 1(+ + ), (5.11a)
d
dx=
b
(1/2)2(C1/C2)2 + 1(+ + ). (5.11b)
Comparing these with (5.3a, b) reveals that we need only modify
our denition of theparameter b when a second liner is included, and
thus the optimal value of Mh,1/1is now (
Mh,1
1
)opt
=(Mh/ )opt,sl
(1/2)2(C1/C2)2 + 1, (5.12)
where (Mh/ )opt,sl is the single-liner result given by (5.8).
For small 1/2, the innerliner is absorbing the larger fraction of
energy. As 1/2 grows the absorption load istransferred to the outer
liner, and when 1/2 = C2/C1, the liners share the load eq-ually.
Note that, because C2/C1 > 1, identical inner and outer liners
do not absorbequally.
Note that expression (5.12) does not provide guidance on how to
choose the ratiosC1/C2 or 1/2. Our simple long-wavelength approach
only states that, provided theparameters are chosen to satisfy
(5.12), the absorption peaks of the liner system willbe
approximately 83%. Solution of the full model reveals that certain
choices of theseratios do lead to marginally better absorption, but
the improvement is too slight tobe incorporated into the design
process.
It is useful to check these expressions with the experimental
results of 4. For theliner system used, C1L/Sp = 5.58, C2/C1 =
1.20, 1 = 0.040 and 2 = 0.020. Thus,equation (5.8) predicts (Mh/
)opt,sl = 1.97 and relation (5.12) produces (Mh,1/1)opt =0.52. The
optimal aperture Mach number, Mh,1 = 0.021, compares favourably
withthe observed peak in the experimental results of gure 9, which
occurs at Mh,1 0.015for the three peak frequencies. Furthermore,
the predicted absorption of 83% is veryclose to the observed values
at the peaks in gure 9.
5.2. Positions of absorption extrema
In the previous section we used the argument that the
long-wavelength limit of thesemi-innite duct solution could be used
for deriving the optimal parameters. We nowexplore whether the same
limit can be used to predict the positions of
absorptionextrema.
Expressions (5.6a, b) indicate that the minima and maxima of the
nite-ductreection coincide with those of the transmission at low
frequency, at values ofkL determined by R = (2n 1) and R = 2n,
respectively. Consequently, theabsorption maxima are quite strong
at low frequency, as indeed are the absorptionminima. Our
approximate expression for the absorption predicts that it will
vanishentirely at the minima. Though this is not true in practice,
the minimum valuesobserved in the experimental results of 4 are
indeed quite small.
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 329
1.2
0.8
0.4
0 0.4 0.6 1.0x/L
L
iner
ene
rgy
flux
0.2 0.8
1.6
(a)
1.2
0.8
0.4
0 0.4 0.6 1.0x/L
0.2 0.8
1.6
(b)
Figure 11. Acoustic energy ux through the liner for a
single-liner system at Mh,1 = 0.04at (a) kL = 0.60 and (b) kL =
0.98. Ld/L = 4.5: ; Ld/L = : .
However, whilst the long-wavelength limit successfully predicts
the value ofmaximum absorption at nite frequencies when the
parameters are optimal (ascomparison with gure 9 demonstrated), the
values of kL at which the absorptionextrema occur are poorly
predicted by R = (2n 1) and R = 2n. As frequencyincreases, the
phases of the semi-innite transmission and reection amplitudes
bothdrift from their long-wavelength values. Consequently, the
phase of Rd
(0) deviatesfrom R , and the peaks and troughs of transmission
and reection tend to separate.Instead of relying on the same
limiting approach for predicting the frequencies ofabsorption
extrema, we will develop semi-empirical relations.
The acoustic energy ux through the liner provides insight into
the absorptionminima. Figure 11(a) depicts (the negative of) the
liner energy ux for a single-linersystem with mean bias ow Mh,1 =
0.040 at kL = 0.60, which corresponds toan absorption minimum. The
results for both nite (open-ended) and semi-innitedownstream ducts
are shown. The plot reveals that for the nite-length case,
theenergy ux through the liner is negligibly small at a position
approximately 54% ofthe liner length. The total absorption, which
corresponds to the area under the linerux curve, is consequently
small.
A node in the stagnation enthalpy at this position of minimum ux
prevents energyfrom travelling through the liner. The presence of
the node is made possible by thedownstream reection, which leads to
a standing wave supported by the downstreamsection as well as a
latter portion of the lined section. In fact, the frequencies
ofminimum absorption can be predicted approximately by using this
standing-waveargument. For certain ranges of frequencies, the
standing wave will have a node inthe lined section. The node is not
complete because liner absorption prevents totalinterference, and
it becomes weaker as it moves toward the upstream end of
thesection. Minimum absorption occurs when the inuence of this node
has the greatestaxial extent, which occurs when the node is located
at some fractional distance from the downstream end of the lined
section. For example, in gure 11(a), 0.46.Since the open duct
ensures a node at the duct exit, frequencies at which multiplesof
half-wavelengths are supported in the section of scaled length Ld/L
+ + /L,where /L is a correction length to account for absorption,
approximately coincidewith absorption minima, predicted by
(kL)(n)min =n
Ld/L + min + min/L. (5.13)
-
330 J. D. Eldredge and A. P. Dowling
From an analysis of results for several values of Ld/L and Mh,1,
in both single- anddouble-liner systems, we have found that
when
min 0.46, min/L 0.05, (5.14)the frequencies are predicted well
by (5.13).
The standing-wave argument does not apply as neatly to
predicting absorptionmaxima. The liner energy ux does not attain a
relative maximum in the lined sectionat maximum absorption as it
does a relative minimum for minimum absorption.Rather, as gure
11(b) shows, the energy ux is maximum at the liner entrance
anddecays steadily with distance as energy is absorbed. Thus, it is
dicult to arguefor odd multiples of quarter-waves in some duct
length coinciding with absorptionmaxima. However, the results for
several sets of parameters can be reduced to asimilar relation to
(5.13), again for an open-ended duct,
(kL)(n)max =(n 1/2)
Ld/L + max + max/L, (5.15)
where, maintaining the same correction length as above,
max 0.60, max/L 0.05. (5.16)It is important to note that the
form assumed for the extrema positions will not hold
in all circumstances. When the downstream duct is longer than
the liner, the approachpredicts the positions well. However, as Ld
approaches L or becomes smaller, theformulae become increasingly
sensitive to and /L, which must ultimately includetheir own
frequency dependences.
5.3. Alternative denition of absorption
Equation (3.8) is not the only denition of absorption, and in
some cases there areother denitions more appropriate for assessing
the performance of an absorbingdevice. An absorption system is
often used to prevent acoustic waves in a ductfrom returning to
their source, and thus mitigate the growth of acoustically
driveninstabilities. The acoustic energy that is transmitted
downstream of the liner sectionis inconsequential, particularly for
a downstream section in which the bulk of thistransmitted energy is
reected by the exit. In these circumstances, when |Rd | 1,equation
(3.8) reduces to
|B+u |2 |Bu |2
|B+u |2 + |B+d |2 . (5.17)An alternative denition for the
absorption is
R 1 |Bu |2
|B+u |2 , (5.18)which is eectively a renormalization of the
traditional absorption of (5.17). This newdenition represents the
decit in the energy reected upstream. For example, for anopen-ended
duct in the absence of any absorption system, nearly all energy
travellingaway from the source is reected back toward it, and R 0.
In contrast, a value of1 for R would indicate that all outgoing
energy is either absorbed or transmitted.
Figure 12 compares the two denitions of absorption using the
model results in atwo-liner system for the case of Mh,1 = 0.023 and
Mu = 0. The corresponding experi-mentally measured values of R are
included, as well. Both absorption denitionsoscillate with
frequency, with extrema at the same frequencies. The absorption
peaks
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 331
0.6
0.4
0.2
0 0.5 1.0 1.5kL
D, D
R
2.0
0.8
1.0
Figure 12. Comparison of absorption denitions for a double-liner
system withMh,1 = 0.023 and Mu = 0. , model: ; R , model: ; R ,
experiment: .
1.0
0.5
0 0.5 1.0 1.5kL
DR
2.0
1.0
0.5
0
DR
1.0
0.5
0
DR
(a)
(b)
(c)
Figure 13. Evaluation of liner performance when varying Ld/L
with Mh,1 = 0.023 andMu = 0. (a) Ld/L = 4.5, (b) Ld/L = 2.25, and
(c) Ld/L = 1.13.
of R rise to 100%, however. Moreover, the absorption troughs are
higher and becomemuch shallower with increasing frequency (the low
experimental value at kL 1.8was attributed in 4.2 to a sensitive
dependence on left/right wave cancellation).
This liner design is an example of an eective absorption system,
but only very nearthe discrete frequencies of peak absorption. It
is often the case that the frequenciesof acoustic waves that should
be suppressed are not known a priori, and thusit is undesirable for
an absorption system to have such narrow bandwidths ofeectiveness.
We can use the analysis of 5.2 to improve the design by spreading
theabsorption minima apart. Equation (5.13) suggests that this can
be accomplished bydecreasing the downstream duct length, Ld/L. In
gure 13 the eect of varying Ld/L
-
332 J. D. Eldredge and A. P. Dowling
is demonstrated. As Ld/L is rst halved and then quartered, the
absorption troughsare more separated and made more shallow.
Furthermore, the absorption peaks arewidened and maintained at
100%.
The extrema formulae, (5.13) and (5.15), can also be used
eectively for predictingthe minima and maxima of the alternative
absorption, provided that kL is sucientlysmall so that the reection
and transmission extrema are close, as discussed in 5.2. Itis
useful to validate the formulae with the model results of gure 13.
For Ld/L = 4.5,the minima are predicted to lie at kL = 0.63, 1.25
and 1.88, and the maxima atkL = 0.31, 0.92, and 1.53. These values
are all near the observed values, thoughthey tend to slightly
overshoot the larger frequencies. At Ld/L = 2.25, the
predictedminimum at kL = 1.14 and maxima at kL = 0.54 and 1.62
agree reasonably well, butwith more overshoot at smaller kL than in
the previous case. Finally, at Ld/L = 1.13,the rst maximum appears
near its expected position at kL = 0.88, but the rstminimum occurs
at a signicantly smaller frequency than the predicted position ofkL
= 1.92. The usefulness of the formulae is limited at this small
Ld/L.
6. ConclusionsIn this work we have investigated the eectiveness
of a perforated liner system
with mean bias ow in the absorption of planar acoustic waves in
a circular duct.The lined section of duct is composed of a
homogeneous array of circular apertures,surrounded by a radially
external region of larger mean pressure that forces a steadyow
through each aperture. Planar sound waves produced by an upstream
sourcetravel through the lined section, subjecting each aperture to
a harmonic pressuredierence that causes the periodic shedding of
vorticity from the aperture rim. Themechanism for absorption lies
in this one-way transfer of acoustic energy into vorticalenergy,
which is subsequently dissipated. Moreover, the fraction of
incident acousticenergy that is absorbed is independent of the
magnitude of the pressure dierence.This linear mechanism is
embodied in the aperture Rayleigh conductivity derived byHowe
(1979b). In this work, we have developed a one-dimensional duct
model witha homogeneous liner compliance based on this Rayleigh
conductivity. The model iscapable of describing both single- and
double-liner congurations.
The comparison of this model with our experimental results
demonstrates excellentagreement for a large range of frequencies
and mean ow conditions. We have shownthat such a system, when
included in a duct whose termination allows most acousticenergy to
reect upstream for further interaction with the liner, can absorb
as muchas 83% of incident energy at certain frequencies, and
prevent 100% of the outgoingenergy from reecting back to the
source. Our analysis of the acoustic energy ux hasrevealed the
local absorptive character of the liner. In particular, it has
demonstratedthat both liners in a double-liner system are important
for absorption. Also, wehave shown that the absorption troughs that
occur at frequencies between the peaksare due to pressure nodes in
the liner that limit the local absorption. The eect ofthese nodes
is crucially dependent on the length of the downstream duct. As
thislength is shortened, the absorption troughs separate with
respect to frequency andthe absorption peaks consequently become
broader.
Further insight has been gained by reducing the model equations
through a long-wavelength limit kL 0 that identied the parameter b
= 1
4(CL/4Sp)(/Mh) as
particularly important in determining the eectiveness of the
liner system as anabsorber. As we have shown, these reduced
equations contain an absorption termthat can be interpreted as a
loss in dynamic head of the steady jets issuing from
-
The absorption of axial acoustic waves by a perforated liner
with bias ow 333
each aperture in the liner. Provided that the ow conditions and
geometry of a singleliner are designed so that b 1/2, the
absorption peaks will be nearly maximal.Furthermore, we have shown
that a double-liner conguration is equally eectivewhen designed
using a modied expression for b that accounts for the
absorptionload shared by each liner.
The mechanism for absorption implicit in the model is the same
as that included inthe reection coecient of Hughes & Dowling
(1990), who also adapted the Rayleighconductivity of Howe (1979b).
The absorption of impinging acoustic waves consideredby Hughes
& Dowling (1990) depends on proximity to a Helmholtz
resonancecondition, involving the imaginary cavity behind each
aperture. Absorption in thepresent model, however, is identied with
the cumulative eect of local absorptionover a duct section of nite
length. The present work involves combining these twoapproaches to
predict the absorption of both axial and radial acoustic modes.
It should be noted that, though we have only considered planar
duct modes in thiswork, the liner system will also absorb
circumferential modes by the same mechanism.A similar analysis is
currently being undertaken to quantify its eectiveness in
thisrespect. Also, the model equations can be adapted to
incorporate combustion byallowing a mean temperature dierence
across the inner liner. Hughes & Dowling(1990) included an
analysis that assumed that, in the presence of high
combustiontemperatures, the mean bias ow creates a thin layer
adjacent to the liner with coldproperties. In the present case,
acoustic waves only travel parallel to this layer ratherthan
impinging upon it, so the eect of combustion would be limited to
modifyingthe speed of sound and mean density in the duct.
This work was supported by the PRECCINSTA project, Contract No.
ENK5-CT2000-00099, which was sponsored by the Fifth European
Community FrameworkProgramme. The authors are indebted to Dr Iain
Dupe`re for many useful discussions.We would also like to thank Mr
Dave Martin for building the experimental apparatus.
Appendix. Analytical solution of single-liner equationsIn this
Appendix we solve the liner equations (2.14a, b) for the special
case of a
single liner exposed to the ambient environment (Conguration 1).
We will assumethat there is no mean upstream duct ow, uu = 0, and
that the liner bias ow issuciently small that ud 0. Thus, the liner
equations reduce to the system
d+
dx=
(ikL +
2
ikL
)+
2
ikL, (A 1a)
d
dx=
(ikL +
2
ikL
) +
2
ikL+, (A 1b)
where CL/Sp , and the numerical liner subscript has been removed
for brevity.The boundary conditions are
+(0) = 1, (A 2a)
(1) Rd+(1) = 0. (A 2b)The factor Rd is a modied reection
coecient, related to the previously denedcoecient by Rd = Rd
exp(i2kL). We have assumed that all acoustic quantities arescaled
by the incident wave amplitude, B+u .
-
334 J. D. Eldredge and A. P. Dowling
The eigenvalues of this system of two rst-order linear equations
are given by
= (k2L2 + )1/2. (A 3)The expression under the square root is
complex and to avoid messy formulae theeigenvalues are left
unevaluated. The solution of the system is
+(x) =[(2 k2L2) + Rd(2 + k2L2)] sinh (1 x) + i2kL cosh (1 x)
[(2 k2L2) + Rd(2 + k2L2)] sinh + i2kL cosh , (A 4a)
(x) =[(2 + k2L2) + Rd(
2 k2L2)] sinh (1 x) i2kLRd cosh (1 x)[(2 k2L2) + Rd(2 + k2L2)]
sinh + i2kL cosh .
(A 4b)
Equations (A 4a, b) do not readily provide understanding of the
physics in the linedsection. However, it is useful to relate this
solution, for which the reection coecientis general, to the
solution for Rd = 0, which corresponds to a semi-innite
downstreamsection. With some manipulation, it can be shown that
+(0)+(x) = +(0)+(x) + (1)(1 x), (A 5a)+(0)(x) = +(0)(x) + (1)+(1
x), (A 5b)
where the subscript denotes the semi-innite duct solution.
Though the physicalsignicance of these relations is limited, they
provide useful expressions whenevaluated at the ends of the lined
section. When the rst is evaluated at x =1 and thesecond at x =0,
the wave amplitudes of both solutions in the upstream and
down-stream ducts are related:
+(0)+(1) = +(0)+(1) + (1)(0), (A 6a)
+(0)(0) = +(0)(0) + (1)+(1), (A 6b)
providing, in eect, a transfer function from the incoming waves,
+(0) and (1),to the outgoing waves, +(1) and (0). An obvious
application of this transferfunction is for transforming results
for a duct of general end condition into resultsfor a semi-innite
duct.
When we apply the boundary conditions on + and to relations (A
6a, b),rearrange, and take their absolute values, we arrive at
|+(1)|2 = +(1)1 Rd(0)
2
, (A 7a)
|(0)|2 =(0) Rd[()2(0) (+)2(1)]1 Rd(0)
2
. (A 7b)
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