Sound radiation from a lined exhaust duct with lined afterbody
Citation for published version (APA):Demir, A., & Rienstra, S. W. (2010). Sound radiation from a lined exhaust duct with lined afterbody. (CASA-report; Vol. 1030). Technische Universiteit Eindhoven.
Document status and date: Published: 01/01/2010
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Mathematics and Computer Science
CASA-Report 10-30
June 2010
Sound radiation from a lined exhaust
duct with lined afterbody
by
A. Demir, S.W. Rienstra
Centre for Analysis, Scientific computing and Applications
Department of Mathematics and Computer Science
Eindhoven University of Technology
P.O. Box 513
5600 MB Eindhoven, The Netherlands
ISSN: 0926-4507
16th AIAA/CEAS Aeroacoustics Conference, 7-9 June 2010, Stockholm, Sweden
Sound Radiation from a Lined Exhaust Duct
with Lined Afterbody
Ahmet Demir
∗Department of Mathematics, Gebze Institute of Technology
and Sjoerd Rienstra
†Department of Mathematics and Computer Science, Eindhoven University of Technology
Aft fan noise radiating from a lined bypass exhaust duct with lined afterbody is modelled in a way that allows an analytical solution of generalised Wiener-Hopf type. A preliminary set of numerically evaluated examples, based on the experiments by Tester at al. seem to confirm their conclusions that the beneficial effect of a lined afterbody is mainly found without mean flow.
I.
Introduction
A. The technical problem
A
FTfan noise radiating from the bypass exhaust duct is usually attenuated by acoustic lining along the inside ofthe duct only. The unconventional treatment of the afterbody, recently proposed by Tester et al. [1–3] has been shown to provide an additional potential of damping of the order of 1 to 4 dB. This is remarkable, given the fact that most of the sound is supposed to radiate away from, rather than towards, the duct axis. In order to understand the underlying mechanisms, no-flow experiments and flow and no-flow CAA simulations have been performed, as well as comparison with a relatively simple analytical model [4] where only the afterbody is lined.
In the present paper we derive an analytical Wiener-Hopf solution of a more complete version, where both the afterbody and the main duct outer wall is lined. This should enable a comparison between a lined duct with hard walled afterbody, and a lined duct with lined afterbody (with and without mean flow). The complexity of the problem leads to quite involved analysis and laborious formulas, but eventually a numerical evaluation provides first results that seem to confirm the earlier findings of Tester et al.
B. The model problem
The typical advantages of analytic solutions (at least, potentially) like transparency and exactness have to be paid for by considerable simplifications of geometry and physical environment in combination with usually highly intricate solution methods. Nevertheless, the complementary information they provide in the form of parametric dependencies and asymptotic behaviour, compared to fully numerical and experimental approaches, can be extremely useful, while the totally uncorrelated type of errors make them ideal for verification of CAA models. Very successful in this respect are the (so-called) Munt models [5–8], based on the idealised geometry of a semi-infinite duct with wall of vanishing thickness and piece-wise uniform mean flows (originally explored by [9, 10]), solved by means of the Wiener-Hopf (WH) method. The classical configurations, with essentially only a single semi-infinite geometrical element, can be solved by means of a form of standard WH method (although the important effects of the edge singularity are not standard). As soon as more geometrical elements are involved, like a semi-infinite duct together with a semi-infinite afterbody, the solution becomes rapidly more complicated. For example, in [4] a generalisation of the WH method was used for the problems of a lined centerbody and of a lined afterbody, using the so-called “weak factorisation”.
∗Assistant Professor, Department of Mathematics, Gebze Institute of Technology, P.O. Box 141, Gebze, 41400 Kocaeli, Turkey
†Associate Professor, Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB
Eind-hoven, The Netherlands, Senior Member AIAA
Copyright c 2010 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.
Very often a superficially innocent looking change of the model requires a complete re-derivation of the solution. For example, the related generalised Wiener-Hopf solution of the configuration with a centerbody that is finite is more difficult (when the inner duct exhaust plane is buried inside the outer duct [11]) or very much more difficult (when the inner exhaust plane is protruding outside the outer duct [12]). Even if the original method is generic, the details may be too different.
Also in present problem, although only a little different from [4], involves a solution procedure that is much more complicated. Again, Idemen’s method [13–15] of “weak factorisation” is applied, but now the Fourier transformed
boundary value problem leads to a 3× 3 matrix Wiener-Hopf equation. This matrix equation system is decoupled by
the introduction of an infinite sum of poles. The uncoupled scalar equations are solved independently by a standard application of analytical continuation. The final solution includes unknown coefficients which are determined by solving an infinite linear algebraic system numerically. The contribution of the instability wave is separated from the rest of the solution. The asymptotic far field is found by a standard application of the steepest descent method.
We end with some practical examples, based on the data of the experiments of [2], showing the potentials of the present model. A more extended scan through the possible parameter values is planned for the future.
To a large extent we follow the lines of [4], but at certain points the analysis differs in a non-trivial way, so we will present the derivation with details.
II.
Analysis
A. Formulation of the mathematical problem
A geometry is considered which consists of a semi-infinite outer duct and a doubly infinite center body. Duct walls are assumed to be infinitely thin and they occupy the region{r = Rh,−∞ < z < ∞} ∪ {r = Rd,−∞ < z < 0} in
circular cylindrical coordinate system (r, θ, z). The outer duct wall is rigid from outside and lined from inside while the center-body is assumed hard inside the outer duct and soft outside the outer duct. The liner impedances are denoted by Zd(duct) and Zh(afterbody). In the region r > Rd, the ambient flow is uniform and axial with density ρ0, velocity
U0and speed of sound c0. In inner region Rh< r < Rd, there exists a jet which is also uniform and axial with density
ρj, velocity Ujand speed of sound cj. A vortex sheet separates these two different flow at the surface z > 0, r = Rd.
All quantities are made dimensionless by the outer duct radius and ambient flow properties
r, z∼ Rd, U ∼ c0, ρ∼ ρ0, t ∼ Rd/c0. (1)
The velocity potential φ, withv = ∇φ, is used to obtain the acoustic pressure p, velocity v and density ρ via the
z= 0
Z
dZ
hM
j,
U
j,
c
j, ρ
jM
0,
U
0,
c
0, ρ
0 r = 0 r= Rh r= Rdfollowing equations [4] p= − ∂φ ∂t + M0 ∂φ ∂ z , v = ∇φ, ρ = p, r> 1 (2a) p= −D1 ∂φ ∂t + M1 ∂φ ∂ z , v = ∇φ, ρ = pC12, h < r < 1 (2b) (2c) where M0= U0/c0, M1= Uj/c0, C1= c0/cj, D1= ρj/ρ0, h = Rh/Rd. (2d)
From the symmetry of the geometry and the incident wave, the diffracted field will remain with the same azimuthal and time dependencies as the incident wave. It is convenient to write the total field in different regions as:
φ (r, θ, z, t )=
ψ1(r, z) eiωt−imθ, r> 1, −∞ < z < ∞,
ψ2(r, z) eiωt−imθ+ψi(r, z) eiωt−imθ, h < r < 1, −∞ < z < ∞,
(3)
where ω is the dimensionless angular frequency (Helmholtz number) and m is the circumferential order. Time depen-dency and the azimuthal dependepen-dency are taken proportional to eiωtand e−imθ, respectively, and suppressed throughout this paper. The incident field is the mode
ψi(r, z)= Amn9mn(r ) e−iωµ
+
mnz. (4)
Here Amn is the amplitude of the incoming wave (which will be taken unity in the analysis) and 9mn is a linear
combination of Bessel functions which satisfy the boundary conditions on the hard and soft wall:
9mn(r )= Yd(µ+mn) Jm(αmn+ r)− Jd(µ+mn)Ym(αmn+ r) (5)
where
Jd(u)= iD1(1− uM1)2Jm(λ1ω)/Zd+ λ1Jm0(λ1ω),
Yd(u)= iD1(1− uM1)2Ym(λ1ω)/Zd+ λ1Ym0(λ1ω).
(6) Scaled axial wavenumbers µ±mnare the roots of the equation
Yd(µ±mn) Jm0(α±mnh)− Jd(µ±mn)Ym0(α±mnh)= 0. (7)
The signs (+) and (−) indicate right and left running modes, respectively. Axial wave numbers ωµ±mnare defined as
µ±mn=−M1 C12± q C12− (1 − M12C12)(αmn± /ω)2 1− M12C12 , (8) where Im µ+mn≤ 0 and Im µ−mn≥ 0.
B. Derivation of the Wiener-Hopf System
The unknown velocity potentials ψ1(r, z) and ψ2(r, z) satisfy the convected wave equations
∂2 ∂r2 + 1 r ∂ ∂r + ∂2 ∂ z2 − m2 r2 − iω+ M0 ∂ ∂ z 2 ψ1(r, z)= 0, r > 1, (9a) ∂2 ∂r2 + 1 r ∂ ∂r + ∂2 ∂ z2 − m2 r2 − C 2 1 iω+ M1 ∂ ∂ z 2 ψ2(r, z)= 0, h < r < 1. (9b)
A solution to these equations can be found via Fourier transformation along z, such that ψ1(r, z)=
ω 2π
Z
L
A(u)Hm(2)(λ0ωr ) e−iωuz du (10a)
ψ2(r, z)=
ω 2π
Z
L
where L is a suitable inverse Fourier transform integration contour along or near the real axis in the complex u-domain (see [4]). Jm and Ym are the Bessel and Neumann functions of order m, Hm(2) = Jm − iYm is the Hankel function of
the second type. λ0and λ1are square root functions which are defined as
λ0(u)= p (1− uM0)2− u2, Im(λ0)≤ 0, λ1(u)= q C21(1− uM1)2− u2, Im(λ1)≤ 0. (11)
Branch cuts for λ0are taken on the line from 1/(1+ M0) to+∞ and from −∞ to −1/(1 − M0). Similarly for λ1
branch cuts start from C1/(1+ M1C1) to+∞ and from −∞ to −C1/(1− M1C1).
−3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1 1 1+M0 C1 1+C1M1 −C1 1−C1M1 −1 1−M0 −M0 1−M2 0
Figure 2. Sketch of complex u-plane with Fourier contour, branch cuts and axial modal wavenumbers.
As usual in this kind of Wiener-Hopf problems, the analysis will be facilitated by giving frequency ω a small negative imaginary part, written as ω= |ω| e−iδ. This will lead to an infinite strip S in the complex u-plane through the origin, inclined under an angle δ, and just small enough to fit between the branch cuts of λ0and λ1, along which
the Wiener-Hopf equation will be formulated. Eventually δ→ 0 and strip S will merge into the real axis.
Equations for the unknown spectral coefficients A(u), B(u) and C(u) will be obtained below from boundary conditions and relations of continuity.
The outer side of the outer duct wall is rigid, so that ∂
∂rψ1(1, z)= 0, z< 0 (12a)
The Ingard-Myers boundary condition [16, 17] along the lined inside part of the outer duct wall can be written as ∂ ∂r + D1 iω Zd iω+ M1 ∂ ∂ z 2 ψ2(1, z)= 0, −∞ < z < 0. (12b)
The same condition along the lined afterbody can be written as ∂ ∂r − D1 iω Zh iω+ M1 ∂ ∂ z 2 ψ2(h, z)+ ψi(h, z)= 0, 0 < z <∞ (12c)
The inner duct wall inside the duct is rigid, so that ∂
∂rψ2(h, z)= 0, z< 0 (12d)
The pressure is continuous across the vortex sheet along r = 1, downstream the trailing edge at z = 0, so D1 iω+ M1 ∂ ∂ z [ψ2(1, z)+ ψi(1, z)]= iω+ M0 ∂ ∂ z ψ1(1, z), z> 0. (12e)
By defining the complex radial displacement of the vortex sheet by r = 1 + ξ(z) eiωt−imθ, we obtain iω+ M0 ∂ ∂ z ξ(z)= ∂ ∂rψ1(1, z), z> 0, (12f) iω+ M1 ∂ ∂ z ξ(z)= ∂ ∂rψ2(1, z)+ ψi(1, z), z> 0. (12g)
which implies the condition of continuity of particle displacement.
In addition to these boundary conditions and continuity relations, we assume that the field radiates outward to infinity and does not reflect backward. A generalized Kutta condition, defining via a parameter 0 the amount of vorticity shed from the cylinder trailing edge, is also imposed at the edge of the cylinder. The full Kutta condition 0 = 1 implies that the pressure is finite at the edge, and the velocity potential is finite and behaves, similar to the vortex sheet displacement, like
ξ(z), φ (1, z)= O(z3/2) z↓ 0 (13)
We introduce the half-plane analytical functions 8±1, F±, G±
8+(u)= Z ∞ 0 ∂ ∂rψ2(h, z) e iωuz dz, (14a) 8−(u)= Z 0 −∞ h∂ ∂rψ2(h, z)− D1 iω Zh iω+ M1 ∂ ∂ z 2 ψ2(h, z) i eiωuz dz, (14b) F+(u)= Z ∞ 0 ξ(z) eiωuz dz, (14c) F−(u)= Z 0 −∞ ∂ ∂rψ2(1, z) e iωuz dz, (14d) G+(u)= Z ∞ 0 h∂ ∂rψ2(1, z)+ D1 iω Zd iω+ M1 ∂ ∂ z 2 ψ2(1, z) i eiωuz dz. (14e) G−(u)= Z 0 −∞ iω+ M0 ∂ ∂ z ψ1(1, z)− D1 iω+ M1 ∂ ∂ z ψ2(1, z) eiωuz dz (14f)
During the course of analysis we will use the fact that the behaviour of a function H+(u) = R0∞h(z) eiωuz dz for |u| → ∞ is related to the (anticipated) behaviour of h(z) for z ↓ 0 in the following way [18]. If h(z) = O(zη), then
H+(u)= O(u−η−1).
Application of the boundary conditions on r = h and r = 1 gives
λ1ω B(u) Jm0(λ1ωh)+ C(u)Ym0(λ1ωh)= 8+(u) (15a)
ω B(u)Jh(u)+ C(u)Yh(u)= 8−(u)− D19mn(h)
(1− M1µ+mn)2
Zh(u− µ+mn)
, (15b)
λ0A(u)Hm(2)0(λ0ω)= i(1 − uM0)F+(u), (15c)
λ1ω B(u) Jm0(λ1ω)+ C(u)Ym0(λ1ω)= iω(1 − uM1)F+(u)+ F−(u)+
9mn0 (1) iω(u− µ+mn)
(15d)
ω B(u)Jd(u)+ C(u)Yd(u)= G+(u), (15e)
iω(1− uM0) A(u)Hm(2)(λ0ω) . . .
−iωD1(1− uM1) B(u) Jm(λ1ω)+ C(u)Ym(λ1ω)= G−(u)− D19mn(1)
1− µ+mnM1 u− µ+mn . (15f) where Jh(u)= −iD1(1− uM1)2Jm(λ1ωh)/Zh+ λ1Jm0(λ1ωh), Yh(u)= −iD1(1− uM1)2Ym(λ1ωh)/Zh+ λ1Ym0(λ1ωh). (16) A(u), B(u) and C(u) may be eliminated as follows
A(u)= i 1− uM0
λ0Hm(2)0(λ0ω)
B(u)= Yd(u)8+(u)− λ1Y
0
m(λ1ωh)G+(u)
λ1ω Yd(u) Jm0(λ1ωh)− Jd(u)Ym0(λ1ωh) ,
(17b)
C(u)= − Jd(u)8+(u)− λ1J
0 m(λ1ωh)G+(u) λ1ω Yd(u) Jm0(λ1ωh)− Jd(u)Ym0(λ1ωh) , (17c) leading to 1 λ1
Jd(u)Yh(u)− Yd(u)Jh(u)
Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh) 8+(u)+Jh(u)Y 0 m(λ1ωh)− Yh(u) Jm0(λ1ωh) Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh) G+= 8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) , (18a) Jd(u)Ym0(λ1ω)− Yd(u) Jm0(λ1ω) Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh) 8+(u)+ λ1 Jm0(λ1ω)Ym0(λ1ωh)− Ym0(λ1ω) Jm0(λ1ωh) Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh) G+(u)=
iω(1− uM1)F+(u)+ F−(u)+
9mn0 (1) iω(u− µ+mn) , (18b) −iD1(1−uM1) 1 λ1 Jd(u)Ym(λ1ω)− Yd(u) Jm(λ1ω) Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh) 8+(u)+ Jm(λ1ω)Y 0 m(λ1ωh)− Ym(λ1ω) Jm0(λ1ωh) Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh) G+(u) − ω(1 − uM0)2 Hm(2)(λ0ω) λ0Hm(2)0(λ0ω) F+(u)= G−(u)− D19mn(1) 1− µ+mnM1 u− µ+mn . (18c)
We define the kernels N (u), L(u), K (u) by N(u)= λ1
Jd(u)Ym0(λ1ωh)− Yd(u) Jm0(λ1ωh)
Jd(u)Yh(u)− Yd(u)Jh(u)
, (19a)
L(u)= λ1
Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
Jd(u)Yh(u)− Yd(u)Jh(u)
, (19b)
K(u)= D1(1− uM1)2
Jm(λ1ω)Yh(u)− Ym(λ1ω)Jh(u)
λ1 Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
− (1 − uM0)2
Hm(2)(λ0ω)
λ0Hm(2)0(λ0ω)
, (19c)
and note the following Wronskian-type relations
Jd(u)Ym(λ1ω)− Yd(u) Jm(λ1ω)= − 2 π ω, Jd(u)Ym0(λ1ω)− Yd(u) Jm0(λ1ω)= 2i D1(1− uM1)2 Zdπ λ1ω , Jh(u)Ym(λ1ωh)− Yh(u) Jm(λ1ωh)= − 2 π ωh, Jh(u)Ym0(λ1ωh)− Yh(u) Jm0(λ1ωh)= − 2i D1(1− uM1)2 Zhπ λ1ωh . (20)
Then equation (18a) can be recast into
8+(u)− N(u) 8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) = 2i D1(1− uM1)2
π ωh ZhJd(u)Yh(u)− Yd(u)Jh(u) G+
(u), (21a)
8+can be eliminated from equation (18b), leading to L(u)G+(u)− iω(1 − uM1)F+(u)= F−(u)+
9mn0 (1) iω(u− µ+mn)
− 2i D1(1− uM1)2
π ω ZdJd(u)Yh(u)− Yd(u)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) , (21b)
By eliminating 8+and G+we find finally from equation (18c)
ωK (u)F+(u)+ 2i D1(1− uM1)
π λ1ω Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) = G−(u)+ i D1(1− uM1)
Jm(λ1ω)Yh(u)− Ym(λ1ω)Jh(u)
λ1 Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
F−(u)+ 9 0 mn(1) iω(u− µ+mn) − D19mn(1) 1− µ+mnM1 u− µ+mn . (21c)
These are the three coupled Wiener-Hopf equations to be solved. Note
We note in passing that we can eliminate G+from equations (21a) and (21b) to obtain
8+(u)− eN(u) 8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) = 2i D1(1− uM1)2
π λ1ωh Zh Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
iω(1− uM1)F+(u)+ F−(u)+
9mn0 (1) iω(u− µ+mn) , with e N(u)= λ1 Jm0(λ1ω)Ym0(λ1ωh)− Ym0(λ1ω) Jm0(λ1ωh)
Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
For a hard-walled duct, F−(u) and 9mn0 (1) vanish, and the expression is equivalent to (28a) and (28b) of [4]. In the present problem on the other hand, the presence of F−apparently prohibits finding a split of the equations in the way done in [4], and we have to retain the set of 3 equations.
C. Factorisation of the kernels
The crucial step in the Wiener-Hopf method is to split the kernel functions N , L and K as a ratio of two functions analytic in the upper and lower half complex u-plane,
N(u)= N+(u) N−(u), L(u)= L+(u) L−(u), K(u)= K+(u) K−(u). (22)
For the meromorphic functions N and L this is relatively easy. For K some care is required because of a physical argument with respect to an instability pole u= u0.
By taking a closed contour withing the strip (see [5,19]) and using Cauchy’s theorem, N±and L±can be evaluated in the classical way by the following integrals [18]
log N±(ξ )= 1 2π i Z C± log[N(u)] u− ξ du, log L±(ξ )= 1 2π i Z C± log[L(u)] u− ξ du. (23)
When δ is taken to zero, the contours C±coincide with the real axis at the respective sides of the branch cuts of λ0
and λ1.
The behaviour for|u| → ∞ is given by
N+(u), L+(u)∼ u−1/2, N−(u), L−(u)∼ u1/2.
In the same way as presented in [19], the zero u = u0of K (u), found in the complex upper half plane, but being
associated to the right-running Helmholtz instability of the vortex sheet really belonging to the the lower half plane, is brought to the “other side” by producing regular split functions ˆK+and ˆK−and then keeping
K+(u)= ˆK+(u)(u− u0), K−(u)= ˆK−(u)(u− u0)
together. Since ˆK+(u)∼ u1/2and ˆK−(u)∼ u−1/2, we have
K+(u)∼ u3/2, K−(u)∼ u1/2. The role of u0is very important for the application of the Kutta condition (see [4]).
−3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1
u
0Figure 3. Sketch of deformed integration contour to include instability pole u0.
D. Wiener-Hopf Solution by Weak Factorisation
Multiplying equation (21a) by 1/N+(u) and subtracting from both sides the pole in u= µ+mnyields 8+(u)
N+(u) −
2i D1(1− uM1)2
π ωh ZhJd(u)Yh(u)− Yd(u)Jh(u)
G+(u) N+(u) + D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(u− µ+mn) = 1 N−(u) 8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) + D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(u− µ+mn) (24a) Multiplying equation (21b) by L−(u) and noting that the poles in u= µ+mncancel out, we obtain
L+(u)G+(u)− iω(1 − uM1)F+(u)
L+(u) λ1
Jd(u)Yh(u)− Yd(u)Jh(u)
Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u) = F−
(u)L−(u)+ 9mn0 (1)
iω(u− µ+mn)
L−(u)− 2i D1(1− uM1)
2
π ω ZdJd(u)Yh(u)− Yd(u)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) L−(u). (24b)
Multiplying equation (21c) by K−(u) and again noting that the poles in u= µ+mncancel out, we get
ωK+(u)F+(u)= G−(u)K−(u)− D19mn(1)
1− µ+mnM1
u− µ+mn
K−(u)
− 2i D1(1− uM1)
π λ1ω Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) K−(u) + iD1(1− uM1)
Jm(λ1ω)Yh(u)− Ym(λ1ω)Jh(u)
λ1 Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
F−(u)+ 9 0 mn(1) iω(u− µ+mn) K−(u). (24c)
Our goal is that the Wiener-Hopf equations (21a,21b,21c) are rewritten in such a way that we have on the left hand side a function that is analytic in the upper half u-plane, and on the right hand side analytic in the lower half u-plane, while both half planes have a strip in common as long as δ > 0. This is almost but not yet achieved by the equations (24a,24b,24c).
Take for example equation (24a). The right-hand side is analytic on the lower half-plane but the analyticity of the left hand side is violated by the zeros u= σmp− of the denominator at the upper half-plane, given by
Jd(σmp−)Yh(σmp−)− Yd(σmp−)Jh(σmp−)= 0.
The method of Weak Factorization involves subtracting the residue contributions from these poles, such that the left hand side becomes analytic in the upper half plane, as follows
8+(u)
N+(u) −
2i D1(1− uM1)2
π ωh ZhJd(u)Yh(u)− Yd(u)Jh(u)
G+(u) N+(u) + D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(u− µ+mn) −X∞ p=1 amp+ u− σmp− = 1 N−(u) 8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) + D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(u− µ+mn) − ∞ X p=1 amp+ u− σmp− (25a) where a+mp= −2iD1(1− σ − mpM1)2
π ωh ZhdudJd(u)Yh(u)− Yd(u)Jh(u)
u=σmp−
G+(σmp−) N+(σmp−)
. (25b)
From Liouville’s theorem it follows that the left and right hand sides define the same analytic function. Anticipating smooth enough behaviour at z= 0 this function is zero, leading to
8+(u)
N+(u) −
2i D1(1− uM1)2
π ωh ZhJd(u)Yh(u)− Yd(u)Jh(u)
G+(u) N+(u) = ∞ X p=1 a+mp u− σmp− − D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(u− µ+mn) , (25c) 1 N−(u) 8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) =X∞ p=1 a+mp u− σmp− − D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(u− µ+mn) . (25d)
Next we consider equation (24b). Here, the analyticity is violated by the zeros u= σmp+ in the upper, and u= κmp− in the lower half plane, given by
Jd(σmp+)Yh(σmp+)− Yd(σmp+)Jh(σmp+)= 0,
Jm0(βmp− )Yh(κmp−)− Ym0(βmp− )Jh(κmp−)= 0, where βmp− = λ1(κmp−)ω.
(Note that λ1 = 0 is not among the poles.) By subtracting the respective poles on both sides, we obtain again
expressions, regular in upper and lower plane:
L+(u)G+(u)− iω(1 − uM1)F+(u)
L+(u) λ1
Jd(u)Yh(u)− Yd(u)Jh(u)
Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)− ∞ X p=1 c+mp u− κmp− − ∞ X p=1 amp− u− σmp+ = − 2i D1(1− uM1)2L−(u)
π ω ZdJd(u)Yh(u)− Yd(u)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) − ∞ X p=1 c+mp u− κmp− − ∞ X p=1 a−mp u− σmp+ + L−(u) F−(u)+ 9 0 mn(1) iω(u− µ+mn) , (26a) where c+mp= 2ω D1(1− κ − mpM1)3F+(κmp− )L+(κmp− )Jh(κmp− )
π Zdβmp− Jm0(βmn− )dud [Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)]u=κmp−
, (26b)
a−mp= − 2i D1(1− σ
+
mpM1)2L−(σmp+)
π ω Zddud Jd(u)Yh(u)− Yd(u)Jh(u)
u=σmp+ ( 8−(σmp+)− D19mn(h) (1− µ+mnM1)2 Zh(σmp+ − µ+mn) ) . (26c)
and use is made of the relation
Jd(κmn− )Yh(κmn− )− Yd(κmn− )Jh(κmn− )= 2i D1 π Zdβmn− (1− κmn− M1)2 Jh(κmn− ) Jm0(βmn−) .
From Liouville’s theorem it follows that the left and right hand sides define the same analytic function. Anticipating smooth enough behaviour at z= 0 this function is zero, leading to
L+(u)G+(u)− iω(1 − uM1)F+(u)
L+(u) λ1
Jd(u)Yh(u)− Yd(u)Jh(u)
Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u) = ∞ X p=1 amp− u− σmp+ + ∞ X p=1 c+mp u− κmp− , (26d)
L−(u) F−(u)+ 9 0 mn(1) iω(u− µ+mn) − 2i D1(1− uM1)2L−(u)
π ω ZdJd(u)Yh(u)− Yd(u)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) = ∞ X p=1 amp− u− σmp+ + ∞ X p=1 c+mp u− κmp− . (26e)
Finally we consider equation (24c). Here, the analyticity is violated by the zeros u = κmp+ in the upper half plane, given by
Jm0(βmp+ )Yh(κmp+ )− Ym0(βmp+ )Jh(κmp+ )= 0, where βmp+ = λ1(κmp+ )ω.
Again, we subtract the residue contributions to obtain
ωK+(u)F+(u)− ∞ X p=1 c−mp u− κmp+ = G−(u)K−(u)− D19mn(1) 1− µ+mnM1 u− µ+mn K−(u)− ∞ X p=1 c−mp u− κmp+ − 2i D1(1− uM1)K−(u)
π λ1ω Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) + iD1(1− uM1)
Jm(λ1ω)Yh(u)− Ym(λ1ω)Jh(u)
λ1 Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
F−(u)+ 9 0 mn(1) iω(u− µ+mn) K−(u), (27a) where c−mp= 2i D1(1− κ + mpM1)K−(κmp+ )
πβmp+ dud Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
u=κmp+ × × " ωJh(κmp+ ) βmp+ Jm0(βmp+ ) ( F−(κmp+)+ 9 0 mn(1) iω(κmp+ − µ+mn) ) − 8−(κmp+ )+ D19mn(h) (1− µ+mnM1)2 Zh(κmp+ − µ+mn) # . (27b)
From Liouville’s theorem it follows that the left and right hand sides define the same analytic function. From estimates
K+ ∼ u3/2in combination with F+ ∼ u−1/2, without Kutta condition, and F+ ∼ u−3/2, with Kutta condition, it
transpires that this analytic function is in general a constant, say E , which is zero with Kutta condition or maximum vortex shedding. Noting, furthermore, that without vortex shedding no instability is excited, while F+does not and K+does have a pole in u= u0, we write the constant in the form E= (1 − 0)E0, with
E0= − ∞ X p=1 c−mp u0− κmp+ ,
such that 0= 0 correspond to no vortex shedding and no pole in u0, while 0= 1 corresponds to a full Kutta condition.
This results in a solution of the form
ωK+(u)F+(u)= ∞ X p=1 c−mp κmp+ − u0 u− u0 u− κmp+ − 0 ! (27c)
This determines the auxiliary function F+(u), necessary for the radiated field, and is therefore the main result of this section.
For completeness, we give, from the right-hand side of (27a) also being equal to (1− 0)E0, the relation
G−(u)K−(u)− 2i D1(1− uM1)K−(u)
π λ1ω Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
8−(u)− D19mn(h) (1− µ+mnM1)2 Zh(u− µ+mn) + iD1(1− uM1)
Jm(λ1ω)Yh(u)− Ym(λ1ω)Jh(u)
λ1 Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)
F−(u)+ 9 0 mn(1) iω(u− µ+mn) K−(u)= ∞ X p=1 cmp− κmp+ − u0 u− u0 u− κmp+ − 0 ! + D19mn(1) 1− µ+mnM1 u− µ+mn K−(u) (27d)
E. Determining the coefficients amp± and cmp±
As the coefficient amp+ is expressed in the unknown G−(σmp−), amp− in 8−(σmp+), c+mpin F+(κmp− ), and c−mpin F−(κmp+) and 8−(κmp+), the solution is not yet known until we have found these coefficients. This, fortunately, is a relatively easy task, since we can set up an infinite system of linear equations by evaluating the pertaining equations at the respective values of u that allows elimination of the unknowns.
From expression (25b) for a+mr, (26d) for G−(σmr−), and using the fact that the factor of F+(σmr−) vanishes, we obtain for r= 1, 2, . . . a+mrπ ωh ZhL+(σ − mr)N+(σmr−) −2iD1(1− σmr−M1)2 d
duJd(u)Yh(u)− Yd(u)Jh(u)
u=σmr− = ∞ X p=1 a−mp σmr− − σmp+ +X∞ p=1 cmp+ σmr− − κmp− . (28a)
From expression (26c) for amr−, (25d) for 8−(σmr+), we obtain for r = 1, 2, . . .
amr−
π ω Zddud Jd(u)Yh(u)− Yd(u)Jh(u)
u=σmr+ −2iD1(1− σmr+M1)2L−(σmr+)N−(σmr+) = ∞ X p=1 amp+ σmr+ − σmp− − D19mn(h)(1− µ+mnM1)2 ZhN−(µ+mn)(σmr+ − µ+mn) . (28b)
From expression (26b) for cmr+, (27c) for F+(κmr−), we obtain for r = 1, 2, . . .
c+mrK+(κ − mr) L+(κmr−) π Zdβmr− Jm0(βmr−) 2ω D1(1− κmr−M1)3Jh(κmr−)
βmr− dud [Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)]u=κmr− = ∞ X p=1 c−mp κmp+ − u0 κmr− − u0 κmr− − κmp+ − 0 ! . (28c)
From expressions (27b) for cmr−, (26e) for F−(κmr+) and 8−(κmr+), and the definition of κmr+, we obtain for r = 1, 2, . . .
c−mr L−(κ + mr) K−(κmr+) πβmr+ Jm0(βmr+) 2iω D1(1− κmr+ M1)Jh(κmr+)
βmr+ dud[Jm0(λ1ω)Yh(u)− Ym0(λ1ω)Jh(u)]u=κmr+ = ∞ X p=1 amp− κmr+ − σmp+ +X∞ p=1 c+mp κmr+ − κmp− . (28d)
The infinite sums in the equations converge very rapidly so they can be truncated quickly.
III.
Far Field
From (10b) with (17b) and (17c), the pressure in the ambient mean flow can be expressed as the following integral
p1(r, z)= ω2 2π Z L (1− uM0)2F+(u) Hm(2)(λ0ωr ) λ0Hm(2)0(λ0ω) e−iωuzdu, (29)
where L is the inverse Fourier transform contour.
For the far field (ω√r2+ z2 1) we can use the asymptotic formula for the Hankel function
Hm(2)(λ0ωr )∼
s 2 π λ0ωr
e−i(λ0ωr−12mπ−14π ), (30)
and replace Hm(2)(λ0ωr ) in (29) to get
p1(r, z)∼ ω2 2π Z L (1− uM0)2 F+(u) λ0Hm(2)0(λ0ω) s 2 π λ0ωr
e−i(ωλ0r+ωuz−12mπ−14π )du. (31)
After transforming the free variables
r= % sin ζ, z= % cos ζ q 1− M02, ω= q 1− M02, u =cos(ζ− iτ) − M0 1− M02 , (τ ∈ R), (32)
leading to λ0ω= sin(ζ − iτ) and an integration contour L deformed into a branch of a hyperbola ¯L (taking account
of possible captured pole contributions), we can evaluate the integral by the method of steepest descent (or, which is equivalent here, stationary phase)
p1(r, z)∼ i2 2π e i%M0cos ζ+i12mπ+i14π s 2(1− M02) π % sin ζ Z ¯L (1− uM0)2F+(u)
Hm(2)0( sin(ζ− iτ))√sin(ζ− iτ)
e−i% cosh τdτ, (33)
plus in a downtream arc the contribution from the pole in u0, i.e., the Helmholtz instability. This instability, however,
will be further ignored here as in physical reality it will diffuse away. (Note that its acoustic relevance, in the form of its scattering at the trailing edge, is included.) The major contribution to the integral comes from the vicinity of τ = 0,
−3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1
Figure 4. Sketch of steepest descent contour in complex u-plane.
wheredτd cosh τ = 0. When we write cosh τ = 1 +12τ2+ . . . , we obtain for large %
Z ∞ −∞e −i% cosh τdτ 'Z ∞ −∞e −i%−1 2i%τ2dτ = e−i% Z ∞ −∞e −1
2i%τ2dτ = e−i%e−i 1
4πr π
%. Hence the solution in the far field can be approximated by
p1(r, z)∼ im+1 √ 2π (1− M0cos ζ )2 (1− M02)3/2 F+ cos ζ− M0 1− M02 ! e−i%(1−M0cos ζ ) Hm(2)0( sin ζ )% sin ζ . (34)
After converting the coordinate variables to the regular physical polar coordinates
z= R cos ϕ, r= R sin ϕ, (35) such that %= q R 1− M02 q 1− M02sin2ϕ, sin ζ = q 1− M02sin ϕ q 1− M02sin2ϕ , cos ζ = q cos ϕ 1− M02sin2ϕ , we obtain p1(R, ϕ)∼ Dp(ϕ) R exp −i ω 1− M02R q 1− M02sin2ϕ− M0cos ϕ ! (36) where pressure field directivity Dpis given by
Dp(ϕ)= im+1 √ 2π ω (1− M02)2 (1− M0cos ζ )2 Hm(2)0( sin ζ ) sin ϕ F+ cos ζ− M0 1− M02 ! (37)
IV.
Numerical evaluation
Since log(N (u)), log(L(u)), log(K (u))∼ ± log |u| for |u| → ∞, the integrals (23) for the split functions do not converge normally at infinity, and we have to convert it into a Cauchy Principle Value integral [18, p.42]. We take the limit symmetrically around a suitable real number between the branch points of λ0and λ1that separates the right and
left running modes (this is not always possible, in which case the contribution of the missed poles have to be added as residues; see [7,8]). Without going into every detail, we obtain typically a contour as depicted in figure 5. More details can be found in [19, 20]. An important point to be checked is whether anywhere along the contour K (u) crosses the branch cut of the logarithm function, normally chosen along the negative real axis. In such a case the integrand is not analytic and the split integral is invalid.
−3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1
Figure 5. Sketch of deformed integration contour for splitfunctions in u-plane.
V.
Numerical Examples
A series of examples are numerically evaluated to see the effect of the lined afterbody, with and without jet flow and duct lining. The problem parameters are chosen from the data set of the experiments of De Mercato, Tester and Holland [2].
For zero flow, the outer and jet properties are the same, where outer flow values are used in both regions. The far field values are plotted dimensionally, at a distance 46.0 m away from the exhaust plane. The incident mode amplitude is taken such that the cross-sectional averaged intensity at z= 0 is 1 W/m2.
The geometry parameters that were used are
Rh = 0.1191 m Mj = Uj/cj = 0.5 ρ0 = 1.225 kg/m3
Rd = 0.1985 m ρj = 1.1921 kg/m3 M1 = Uj/c0 = 0.507026
h = Rh/Rd = 0.6 U0 = 0 m/s C1 = c0/cj = 0.986143
Uj = 172.475 m/s c0 = 340.17 m/s D1 = ρj/ρ0 = 0.973143
cj = 344.95 m/s M0 = U0/c0 = 0
all with full Kutta condition, in combination with the following frequencies and impedances
f1 = 1250 Hz Zd (duct) = 1 − 7.27i
ω1 = 2π f1Rd/c0 = 4.58305 Zh (afterbody) = 1 − 7.27i
m = 0 µ+01 (hard) = 0.6574286 µ+01 (soft) = 0.67753 − 0.00275i
f5 = 3150 Hz Zd (duct) = 1 − 2.79i
ω1 = 2π f5Rd/c0 = 11.54928 Zh (afterbody) = 1 − 2.79i
m = 0 µ+01 (hard) = 0.6574286 µ+01 (soft) = 0.67923 − 0.00925i
m = 5 µ+51 (hard) = 0.5043326 µ+51 (soft) = 0.54708 − 0.01718i
f6 = 4000 Hz Zd (duct) = 1 − 2.14i
ω1 = 2π f6Rd/c0 = 14.66575 Zh (afterbody) = 1 − 2.14i
m = 6 µ+61 (hard) = 0.5243871 µ+61 (soft) = 0.56989 − 0.02547i
µ+62 (hard) = 0.3054997 µ+62 (soft) = 0.34767 − 0.01544i
In the following sets of cases the effect of adding a lined afterbody to a hard-walled duct and a lined duct, with and without flow is considered. Note that it is not possible to compare the radiation of the hard-walled duct with the lined duct, because the location of the source, and hence the attained damping, is unknown. Therefore, we will only compare the adding of the afterbody liner with a reference situation of a hard-walled duct and with another reference situation of the lined duct.
A. Frequency f1
The 1st set, given in figure 6, shows the field of the first radial m = 0-mode for a relatively low frequency and a
relatively high impedance. The 2nd set, given in figure 7, shows the corresponding first radial m= 1-mode. Adding
the afterbody liner gives only a slight reduction in all cases. This is probably due to the high impedance used.
B. Frequency f5
The 3d set, given in figure 8, shows the field of the first radial m = 0-mode for a higher frequency and a moderate
impedance. The 4th set, given in figure 9, shows the corresponding first radial m = 5-mode. Adding the afterbody
liner gives a considerable reduction in the downstream arc without flow, which broadly agrees with the findings of [2]. With flow this reduction is much less.
C. Frequency f6
The 5th and 6th sets, given in figures 10 and 11, show the fields of the first and second radial m = 6-modes for a
rather higher frequency and a moderate impedance. Again, adding the afterbody liner gives by and large a reasonable reduction in the downstream arc without flow, while with flow this reduction is much less, and only in the upstream arc.
VI.
Conclusions
Experimental evidence for the no-flow configuration by Tester et al. [1,2] showed that a lined afterbody could have an overall beneficial effect on the radiation field from an exhaust duct by as much as 1 to 4 dB. The presence of mean flow however, addressed by numerical simulation in [3], seemed to limit this reduction.
In order to produce a complementary, fully exact, solution of Wiener-Hopf type that could (in principle) compare these cases, we developed the present model of sound radiation from a semi-infinite duct, with jet flow and co-flow while the duct is lined on the inside of the outer wall, and a doubly infinite hub that is lined on the part outside the duct.
The Wiener-Hopf solution for this kind of problems is non-standard and quite formidable, because the usual factorisation is not possible, and we have to resort to the so-called weak factorisation method.
The formal solution has been implemented numerically, and the results obtained showed (for the limited number of cases considered) by and large a confirmation of the conclusions of Tester et al. More extended parameter scans for more impedance values should be done to see if impedances are possible that retain with flow the interesting attenuations achieved without flow.
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(a) No flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(b) No flow, lined duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(c) With flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(d) With flow, lined duct Figure 6. Frequency f1with m= 0 and the 1st radial mode
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(a) No flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(b) No flow, lined duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(c) With flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(d) With flow, lined duct Figure 7. Frequency f1with m= 1 and the 1st radial mode
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(a) No flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(b) No flow, lined duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(c) With flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(d) With flow, lined duct Figure 8. Frequency f5with m= 0 and the 1st radial mode
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(a) No flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(b) No flow, lined duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(c) With flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(d) With flow, lined duct Figure 9. Frequency f5with m= 5 and the 1st radial mode
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(a) No flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(b) No flow, lined duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(c) With flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(d) With flow, lined duct Figure 10. Frequency f6with m= 6 and the 1st radial mode
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(a) No flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(b) No flow, lined duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o hard Z h
(c) With flow, hard duct
47 54 61 68 75 dB 0o 30o 60o 90o 120o 150o 180o Z d both
(d) With flow, lined duct Figure 11. Frequency f6with m= 6 and the 2nd radial mode
VII.
Acknowledgement
The present work was inspired by, and a continuation of, our work carried out under the European collaborative project “TURNEX” under the Sixth Framework Programme (Technical Officer Daniel Chiron, Project Coordinator Brian Tester). We would like to express our appreciation for the cooperation with the partners of the project.
References
1B.J. Tester, L. De Mercato and K. Holland, Aft fan noise reduction with a lined afterbody. AIAA 2007-3527 of the 13th AIAA/CEAS
Aeroacoustics Conference, 2007
2L. De Mercato, B.J. Tester and K. Holland, Aft fan noise reduction with a lined afterbody. Paper 431 of the 14th ICSV, Cairns, Australia,
2007
3L. De Mercato and B.J. Tester, After-body Liner Performance Predictions. AIAA 2008-2881 of the 14th AIAA/CEAS Aeroacoustics
Con-ference, 2008
4A. Demir and S.W. Rienstra, Sound Radiation from an Annular Duct with Jet Flow and a Lined Centerbody, AIAA 2006-2718 of the 12th
AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, 8-10 May 2006
5R.M. Munt. The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe. Journal of Fluid Mechanics,
83(4):609-640, 1977.
6S.W. Rienstra, Acoustic Radiation From A Semi-Infinite Annular Duct In A Uniform Subsonic Mean Flow, Journal of Sound Vibration,
94(2):267-288, 1984
7G. Gabard, R. Astley and M. Ben Tahar Noise Radiation from a Jet Pipe: A Benchmark Problem for Computational Aeroacoustics
AIAA-2005-3064 11th AIAA/CEAS Aeroacoustics Conference, 23-25 May 2005
8G. Gabard and R.J. Astley, Theoretical models for sound radiation from annular jet pipes: far- and near-field solutions, Journal of Fluid
Mechanics, 549:315-342, 2006
9H. Levine and J. Schwinger. On the radiation of sound from an unflanged circular pipe. Physical Review, 73:383-406, 1948 10G.F. Carrier. Sound transmission from a tube with flow. Quarterly of Applied Mathematics, 13:457-461, 1956.
11A. Demir and S.W. Rienstra, Sound Radiation from a Buried Nozzle with Jet and Bypass Flow, paper no. 449 at ICSV14, the Fourteenth
International Congress on Sound and Vibration, Cairns, July 9-12, 2007.
12B. Veitch and N. Peake, Acoustic propagation and scattering in the exhaust flow from coaxial cylinders, Journal of Fluid Mechanics, 613,
275-307, 2008
13M. Idemen, A new method to obtain exact solutions of vector Wiener-Hopf equations, ZAMM 59:656-658, 1979
14A. Büyükaksoy and A. Demir, Radiation of sound from a semi-infinite rigid duct inserted axially into a larger infinite tube with wall
impedance discontinuity, ZAMM (in press)
15I.D. Abrahams, Scattering of sound by two parallel semi-infinite screens, Wave Motion 9:289-300, 1987
16K.U. Ingard, Influence of Fluid Motion Past a Plane Boundary on Sound Reflection, Absorption, and Transmission, Journal of the Acoustical
Society of America31(7), 1035–1036, 1959
17M.K. Myers, On the acoustic boundary condition in the presence of flow, Journal of Sound and Vibration, 71 (3), p.429–434, 1980 18B. Noble, Methods based on the Wiener-Hopf Technique, Pergamon Press, London, 1958.
19S.W. Rienstra and N. Peake, Modal Scattering at an Impedance Transition in a Lined Flow Duct, AIAA 2005-2852 of the 11th AIAA/CEAS
Aeroacoustics Conference, Monterey, CA, 23-25 May 2005
