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On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow
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On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow Changyong Jiang , Chunqi Wang PII: DOI: Reference:
S0020-7403(19)31710-2 https://doi.org/10.1016/j.ijmecsci.2019.105300 MS 105300
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
15 May 2019 11 October 2019 2 November 2019
Please cite this article as: Changyong Jiang , Chunqi Wang , On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105300
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Highlights Eigenmode analysis of lined flow duct with arbitrary cross section Two kinds of eigenmodes are observed and characterized Mode properties change with frequency and Mach number Mach number affects wavenumber of air-borne mode but mode shape of lining-borne mode
1
On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow Changyong Jianga,b,c, and Chunqi Wangb,c,* a
College of Engineering Physics, Shenzhen Technology University, Shenzhen 518118, China b
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China
c
Laboratory for Aerodynamics and Acoustics, HKU-Zhejiang Institute of Research and Innovation, Lin An 311305, Zhejiang, China *Corresponding author, E-mail: [email protected] Phone: 852-28592621; Fax: 852-28585415
Abstract: The present work concerns the numerical eigenmode analysis of acoustically lined ducts of any constant cross-sectional shape in the presence of mean flow. The duct is lined with bulk-reactive porous material, which is separated from the flow channel by a perforated panel. A combined wave and finite element (WFE) method is introduced to investigate the effects of uniform mean flow on the eigenmodes and eigenvalues of the lined ducts. The proposed method is validated against analytical solutions for a simple case. A convergence rate of Se4 for eigenvalues can be reached for Mach number up to 0.8, with Se being the mesh size. In this study, the eigenmodes of a lined duct are divided into air-borne modes and liningborne modes. The air-borne modes propagate mostly in the air domain, while the lining-borne modes can propagate in both air domain and porous domain. The effect of Mach number on the axial wavenumbers and mode shapes are analysed. It is found that, for air-borne modes, Mach number will strongly affect their wavenumbers, while its effect on the mode shape is marginal. On the contrary, for the lining-borne modes, Mach number has small effect on the wavenumber, but its effect on the mode shape is significant. Specifically, even the mode properties can be changed from a lining-borne mode to an air-borne mode for some left-travelling lining-borne mode. The transmission loss (TL) of a lined duct with three typical Mach numbers is calculated by the mode matching method. It is found that up-flow will generally increase TL of lined duct, compared with the case of no flow; while down-flow will generally decrease TL. In both situations, the peak frequency does not show significant change. Keywords: eigenmode analysis; mean flow; duct lining; eigenvalue problem; finite element method
2
1
Introduction Acoustically lined ducts are commonly used in various engineering applications for noise control
purpose, like heating, ventilation, and air conditioning (HVAC) systems, automobile cars, and aviation industry. In a previous research, a poroelastic lined duct is analyzed based on the calculated crosssectional eigenmodes [1], but the effect of flow was not considered. In real applications like HVAC systems, the air passage generally contains flow, which may affect the lining properties seriously. The present work concerns the numerical eigenmode analysis of acoustically lined ducts of any constant cross-sectional shape in the presence of mean flow. The effect of uniform mean flow on the eigenmodes is focused. In the literature, various methods have been developed to calculate the eigenmodes of acoustic duct linings with mean flow. An extensive review on this subject can be found by Kirby in 2009 [2]. When the cross-section has a simple, regular shape (rectangular, circular or annular), an analytical eigenvalue problem for the duct lining can be formulated [3, 4]. Generally, a robust root-finding algorithm is needed to find solutions of the eigenvalue equation, which is not a trivial task [5]. Recently, many researchers have made new investigations into this topic. Jiang et al. proposed an analytical method to model the lined duct with azimuthally non-uniform impedance [6]. In their modelling, Fourier series expansion was incorporated into the Wiener-Hopf method and uniform mean flow in the duct was considered. Qiu et al. formulated the analytical eigenvalue problem of lined annular duct with uniform mean flow [7]. In their research, the Cremer concept [8] was further investigated to optimize the surface impedance, with which maximum noise reduction for the eigenmode could be achieved. Bravo and Maury investigated sound attenuation of non-locally lining structures by anisotropic fibrous materials in contact with uniform mean flow [9]. The least attenuated mode of the lining system was obtained with a simulated annealing search method. In addition to the aforementioned frequency domain method, time domain method to obtain eigenmodes has also been proposed [10]. For complex cross-sectional shapes, an analytical formulation is usually impossible. It is an attractive alternative to solve the eigenvalue relationship numerically to avoid root-finding, like the point-matching method [11], Chebyshev collocation method [12] and so on, but a regular cross-sectional geometry is a prerequisite. Recently, a semi-analytical formulation was developed to investigate the acoustic eigenproblems of three-dimensional elliptical cylindrical cavities with multiple elliptical cylinders [13]. Having said these, a discrete formulation for the cross-sectional eigenmodes are more preferred with arbitrary geometry, e.g. the finite element (FE) formulation and mesh-free methods. A FE approach is more versatile for any type of cross-sectional geometry. In this regard, Son et al. provided the FE formulation of eigenvalue problem for poroelastic lined duct [14]. In their study, the influence of flow
3
and perforated panel were ignored. Astley and Cummings established the eigenvalue problem with FE formulation for a duct lined with bulk-reactive porous material, while considering the uniform mean flow [15]. Kirby took the influence of perforated panel into consideration, as well as the uniform mean flow [16]. Recently, Denia et al. obtained the eigenvalues and associated eigenmodes for the silencer’s twodimensional (2D) cross section with FE formulation [17]. Effect of uniform mean flow and transverse temperature gradient was included in the model. A point collocation technique was then used to predict transmission loss (TL), which matched favorably well with a full three-dimensional (3D) FE simulation, with reduced computational effort. Sanchez-Orgaz et al. obtained the eigenmode of 2D cross-section of a lined duct through FE formulation, with the effect of perforated panel considered [18]. The granular material was used as sound absorption material, but the effect of flow was not considered. TL was also obtained through mode matching method. Fang and Liu investigated eigenmodes of the silencer crosssection through a semi-weak-form mesh-free method [19]. In their method, shape functions were created with the radial basis function point interpolation method. Galerkin method was adopted to obtain the weak formulation of transversal governing equations, and the Gauss integration scheme was used to compute the system matrices. However, a discrete formulation of the eigenvalue problem by direct FE or mesh-free method requires specific prior knowledge of the application procedure, which is beyond the capability of many engineers. It is therefore desirable to develop an alternative approach that is more efficient to apply. This study seeks to investigate the effect of uniform mean flow on the eigenmode properties of an acoustically lined ducts of any constant cross-sectional shape with a wave and finite element (WFE) method. With this method, the effect of bulk-reactive porous material and perforated panel can be easily taken into account. In addition, the method can be applied to acoustic duct linings with arbitrary crosssectional shape, as long as it is uniform along the axial direction. The general idea of the WFE method was first proposed in [20] to analyze the one-dimensional wave propagation problems. The major advantage of such method is that it formulates the eigenvalue problem based on general FE matrices, which can be obtained conveniently from commercially available FE softwares. Hence, the modelling procedure of even complicated problems becomes relatively easy to handle which is of considerable practical interest. The WFE method has been successfully applied to investigate wave propagation problems in beams [21, 22] and plates [23, 24]. In this paper, the eigenmodes of porous lined duct with uniform mean flow is investigated and analyzed. Different types of modes are identified base on mode properties, and mode evolution with frequency and Mach number is investigated. The rest of the paper is structured as follows. In Sec. 2, the modelling and eigenvalue problem formulation procedure are described. In Sec. 3, the proposed approach is validated with analytical solutions. In Sec. 4, the effect of Mach number on the eigenmodes of the lined duct is presented. Conclusions are drawn in Sec. 5.
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2 2.1
Methodology Problem description In the present study, a porous material lined duct with uniform mean flow in axial direction is
considered, whose cross-section can be arbitrary in shape. Between the air domain and porous lining domain, a perforated panel is used. The axial sectional, cross-sectional and three dimensional (3D) views are shown in Fig. 1(a)-(c), respectively. The acoustic wave in the air domain of the lined duct is described by the convected wave equation, 2 p a
1 D2 pa 0, c02 Dt 2
(1)
where pa is the acoustic pressure for air domain, c0 is the speed of sound in stationary air, D2 D U 0 , and U0 is the mean flow velocity. The calculation is conducted in 2 Dt z Dt t 2
2
frequency domain, where the time dependence of eit is assumed, ω = 2f is angular frequency, f is frequency with the unit of Hz and i 1 . Therefore, Eq. (1) can be reformed as,
2 p a 1 M 2
zp 2
2
a
2ik0 M
p a k02 p a 0 , z
(2)
where 2 is the Laplace operator in the cross-section, M = U0/c0 is the Mach number, k0 = /c0 is the wavenumber of sound in air. The particle displacement in air domain ua can be calculated with Euler equation
0
Du a p a . Dt
(3)
with ρ0 being the air density. For the porous lining domain, the porous material is modelled with the semi-empirical model proposed by Allard and Champoux [25]. In the literature, Delany and Bazely’s empirical model [26] and Johnson-Champoux-Allard’s (JCA) physical model [27, 28] are widely used. However, both of them have some restrictions. Delany model is valid in a limited frequency range, which may cause unphysical results in low-frequency range. Although Miki [29] has made modifications to Delany’s model, problem in lowfrequency is not completely solved [30]. JCA model more accurate with physical bases, which can also be adopted here as in the previous literature [1], but it is more complicated with several parameters which is not easy to determine, like viscous and thermal characteristic lengths. Therefore, the semi-empirical model [25] is adopted in this study, which is simple to implement with one parameter while keeping meaningful results in low-frequency range. With this model, the porous material is treated as an equivalent fluid with its equivalent density ρc and elastic modulus Kc expressed as 5
2 1 1.2 0.0364 0 f i0.1144 0 f Rf Rf c 2 0 f f i29.64+ 2.82 i24.9 0 Rf Rf K c 101320 2 0 f f i21.17+ 2.82 i24.9 0 Rf Rf
1
1
,
(4)
where Rf is the flow resistivity. The equivalent speed of sound cc, characteristic impedance zc and wavenumber kc of porous materials are calculated as: cc K c c . zc c cc k z c c
(5)
The acoustic field in the porous lining domain is governed by the Helmholtz equation with the equivalent fluid model,
2 p p kc2 p p 0 ,
(6)
where pp is the acoustic pressure in porous lining domain. The particle displacement in the porous domain up is then obtained as
up
1
c
2
p p .
(7)
The perforated panel between the air domain and porous lining domain is modelled with the model proposed by Allam and Abom [31]. In this model, the perforated panel is treated as an impedance boundary with impedance zpp z0 rpp xpp i , where z0 = ρ0c0 is the characteristic impedance of air, rpp and xpp are resistance and reactance, respectively, determined as
i i
2 J1 i t h r Re 1 pp c i J0 0 2 J1 i t xpp Im h 1 c0 i J 0
1
i i
1
K pp M uˆh 2 Rs c c 0 0 0 uˆh pp F 1 c0 c0
1
.
(8)
where Re and Im mean taking real part and imaginary part of a complex variable respectively, J0 and J1 mean the zero-th and first order of Bessel function, respectively, σ is perforated ratio of the panel, th is the thickness of the perforated panel, d 4 is a dimensionless variable, relating to the diameter of the 6
holes in the panel d, and kinetic viscosity of air , Rs
1 20 is the surface resistance, η is the 2
dynamic viscosity of air, pp = 8d/3 is the end correction of the hole, uˆh is the averaged particle velocity
3 1
in the hole, Kpp 0.15 0.0125 std represents the effect of flow on resistance, F 1 12.6M represents the effect of flow on reactance.
Note that Eq. (8) is used for air-perforated panel-air system, which is a little bit different from the case described here. Actually, when the porous material is in close contact with the perforated panel, the impedance of the perforated panel can be much more complicated, as investigated by Kirby and Cummings for the case with mean flow [32] and Lee et al. for the case without mean flow [33]. In Kirby and Cummings research [32], it was concluded that the proposed semi-empirical scheme yielded good result for resistance and at least adequate result for the mass end correction. In practice, it is difficult to ensure a perfect close contact between the perforated panel and the porous material. Hence, a small air gap greater than the hole diameter is assumed between the perforated panel and the porous material, as adopted and experimentally validated by Bravo and Maury [9]. In this case, the effect of porous material on the panel impedance can be neglected. If air gap is smaller than the hole diameter, impedance of the perforated panel should be modified [32-34]. In the present study, assumption by Bravo and Maury [9] is used, so the influence of porous material on perforated panel is not taken into consideration. At the perforated panel, which is air-porous domain interface, referring to Fig. 1, the following compatibility conditions are applied,
u a n u p n , a p p p zpp vn
(9)
where n is the normal vector of the interface, vn is the normal component of particle velocity in porous lining domain.
7
Porous material Perforated panel L I R
Air
U0
z axis
(a) Axial-sectional view
y
z axis Air 0
x
Porous material
(b) Cross-sectional view
(c) 3D view
Fig. 1. Geometry of the porous lined duct with arbitrary cross-section containing a uniform mean flow U0; (a) axial-sectional view with axial direction set as z axis; (b) cross-sectional view with 2D coordinate system; (c) 3D view for a short segment with length . 2.2
Eigenvalue problem formulation The eigenvalue problem for the cross-section of the porous lined duct is formulated using WFE
method [1, 20]. In this formulation, the acoustic field on the cross-section forms the eigenmode, and the corresponding axial wavenumber forms the eigenvalue. Therefore, the solved eigenmode is for the 2D cross-section, and the eigenvalue depends on the frequency. The major steps of the procedure are presented below for completeness. A short segment of the lined duct with axial length , which is marked in Fig. 1(a) and depicted in 3D form in Fig. 1(c), is modelled with the commercial finite element (FE) software COMSOL Multiphysics®, and the equation of motion is expressed as
K iC M q f , 2
(10)
where, K, C and M are the stiffness, damping and mass matrices, respectivly; q is nodal degree of T
p freedoms (DOFs); f is nodal forces. In this study, the DOFs are p in porous lining domain and T
p a in air domain, with the superscript T denoting matrix transpose. The porous material is modelled as a common fluid with complex density and complex elastic modulus, and the effect of perforated panel is represented by its impedance. 8
The short segment is meshed with structured quadratic lagrangian elements, where one element is used along the axial direction. Introducing the dynamic stiffness matrix D K iC 2M , the equation of motion can be expressed as Dq f . As shown in Fig. 1(a), the nodes in the short segment can be divided into left cross-sectional (L), internal (I) and right cross-sectional (R) nodes. Dividing the nodal DOFs and corresponding nodal forces into L, I and R parts, the equation of motion is partitioned as, DII DLI DRI
DIL DLL DRL
D IR q I 0 D LR q L f L . D RR q R f R
(11)
Eliminating qI in Eq.(11), the equation of motion for the left and right cross-section DOFs are obtained as,
DLL D RL
DLR q L f L , DRR q R f R
(12)
with the expression of matrix elements DLL, DLR, DRL and DRR provided in Appendix B. Re-ordering Eq. (12), the tranfer matrix formulation from the left cross-section to the right cross-section is obtained
q q T L R , fL fR
(13)
where the elements in matrix T are displayed in Appendix B. Combining Eq. (13) and periodic boundary condition
qR qL , fR fL
(14)
q q T L L . fL fL
(15)
the standard eigenvalue problem is formed as
The eigenvalue corresponds to the axial wavenumber kz of the eigenmode of the porous lined duct, with the relation
eik . z
(16)
The imaginary part of the wavenumber represents the attenuation of corresponding eigenmode, while real part implies the phase angle change. The first half of the eigenvector relates to the mode shape of the eigenmode. The analysis here concerns an infinite lined duct, so the axial wavenumbers are related to waves in a duct of infinite length. Since all the wavenumbers are complex numbers due to damping effect of porous materials, there is no rigorous criterion to determine the propagating, attenuating or evanescent modes. In this case, all the modes are simply divided into right-travelling and left-travelling waves with negative and positive imaginary parts of wavenumbers, respectively. In other words, the propagation 9
direction of the modes is determined by the requirement that all the modes are damped along the propagating path. When the Mach number is positive (flow direction is rightward), right-travelling modes propagate in the same direction as the uniform mean flow, while left-travelling modes propagate in the opposite direction. 3
Convergence analysis of the numerical scheme In this section, the calculated eigenvalues by WFE method are validated and the convergence rate of
eigenvalues is investigated. To compare with analytical results, a relatively simple geometry of a cylindrical duct without porous lining and perforated panel is considered. The radius of the circular crosssection is set as 0.1 m. For a circular duct with uniform flow, the axial wavenumbers of the cross-sectional eigenmodes can be analytically expressed as [35]
k
z ,m,n
Mk0 k02 1 M 2 kr2,m,n 1 M 2
, m, n 0,1, 2
,
(17)
where the superscripts + and − mean right-travelling and left-travelling modes, respectively; the subscripts r and z mean the axial and radial direction, respectively; the subscripts m and n are the mode index for circumferential and radial directions, respectively. kr,m,n is the n-th solution for the following eigen equation
dJ m kr r dr
0 at r r1 ,
(18)
where Jm is the m-th order Bessel function. The corresponding eigenmode Φm,n with the axial wavenumber kz,m,n can be expressed as
m,n r , J m kr ,m,n r eim .
(19)
The eigenmodes are referred to as (m,n) mode, with m and n being integers, where the first integer m represents the circumferential order and the second integer n represents the radial order. The relative error is defined as
kˆ k z
z
k z to show the convergence of the proposed method,
where kz is the value calculated by (17), and kˆz is the result calculated by the WFE method. The duct radius r1 is chosen as 0.1 m, and the calculated frequency in this section is chosen as 4000 Hz, if not specified. In our research, M < 1 is considered, so the calculated Mach number range is set as [0, 0.8].
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3.1
Chosen of segment length The chosen of segment length Δ can affect the accuracy of the WFE method, so its effect is
investigated for different Mach numbers. In this section, the maximum element size is chosen as 1 mm, which will be shown to be good enough for convergence. The reletive error for right-travelling modes from analytical solution with M = 0, 0.5 and 0.8 are displayed in Fig. 2(a)-(c), respectively. It is found that for a fixed Mach number, the reletive error generally decreases with the decrease of Δ and then converges for both lower order modes (e.g. (1,1) mode) and higher order modes (e.g. (5,5) mode). However, compared with lower order modes, a smaller Δ is needed for the convergence of higher order modes. In the calculated example, Δ = 10-4 m can ensure the convergence of both lower order and higher order modes. In addition, the convergence patterns with Mach numbers from M = 0 to M = 0.8 are nearly the same. Therefore, in the following calculation, Δ = 10-4 m is chosen.
10
Relative error
10 10 10 10 10
0
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
-10
(a) M = 0
10-1
10-2
10-3 10-4 Segment length [m]
11
10-5
10
Relative error
10 10 10 10 10
0
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
-10
(b) M = 0.5
10-1
10
Relative error
10 10 10 10 10
10-2
10-3 10-4 Segment length [m]
10-5
0
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
-10
(c) M = 0.8
10-1
10-2
10-3 10-4 Segment length [m]
10-5
Fig. 2. Effect of segment length Δ on wavenumbers (relative error compared with analytical value) for right-travelling (1,1), (3,3) and (5,5) modes; (a) M = 0; (b) M = 0.5; (c) M = 0.8. 3.2
Convergence with mesh size The convergence of calculated wavenumbers with mesh size is investigated. The mesh size Se is
defined as the maximum element size of the mesh. When the mode expansion methods are used, convergence rate can be defined as the convergence with number of modes used [36, 37]. In FE analysis, the convergence with number of total nodes used is often adopted. However, the number of total nodes needed is dependent on the geometry of the calculation domain. Using the mesh size Se defined here, the convergence rate is independent on the geometric parameters, which shows more intrinsic convergence properties of the WFE method. 12
The relative error of calculated axial wavenumbers with the increase of element number are displayed in Fig. 3(a)-(c), respectively for M = 0, 0.5 and 0.8. In addition, a curve for trend of Se4 (relative value with reference value adjusted to show the trend in the figure) is also plotted for reference.
Relative error
10
10
10
10
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
Se4 10
Relative error
10
10
10
10
-10
(a) M = 0 8
4 2 Mesh size Se [mm]
1
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
Se4 10
-10
(b) M = 0.5 8
4 2 Mesh size Se [mm]
13
1
Relative error
10
10
10
10
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
Se4 10
-10
(c) M = 0.8 8
4 2 Mesh size Se [mm]
1
Fig. 3. Convergence of wavenumbers (relative error compared with analytical value) with mesh size Se for right-travelling (1,1), (3,3) and (5,5) modes, together with the trend of Se4 provided for reference; (a) M = 0; (b) M = 0.5; (c) M = 0.8. It is found that the relative error of calculated axial wavenumbers decreases with the decrease of mesh size Se, whose trend is parallel to the trend of Se4. Therefore, a convergence rate of Se4 for both lower order and higher order modes with Mach number ranging from 0 to 0.8 is realized with the WFE method. However, smaller mesh size is needed for higher order modes to achieve the same accuracy as lower order modes. The convergence rate at different frequencies is also investigated. The calculation example for (3,3) right-travelling mode at three different frequencies is shown in Fig. 4. It is found that the convergence rate of Se4 is kept up to 8000 Hz.
14
Fig. 4. Convergence of wavenumber with mesh size Se at three different frequencies, shown as relative error compared to analytical value, for the (3,3) right-travelling mode, together with the trend of Se4 provided for reference. 4
Effects of mean flow on the eigenmode characteristic of the lined ducts The eigenmodes of the porous lined duct with mean flow can be obtained with the WFE method, as
described Section 2. In this section, the cross-sectional geometry of the lined duct is set as circular for easier analysis, as depicted in Fig. 5(b).
Γinlet A U0
B
I
pi
Γ1 z 0 C r
L0
Perforated panel
Γ2 Air
r2
II
F
E L (a) Axial-sectional view
Porous material r1
Air
r2 (b) Cross-sectional view 15
III L0
Porous material D
r1
Γoutlet H G
Fig. 5. Geometry of the porous lined duct with circular cross-section; (a) axial sectional view; (b) crosssectional view. 4.1
Eigenvalue and eigenmode In this section, the eigenmodes and corresponding wavenumbers are calculated for the lined duct,
with the geometry and material parameters described in Table 2. The calculated axial wavenumbers for the eigenmodes of the porous lined duct are shown in Fig. 6. The Mach numbers is set as M = 0.2, the circumferential order is set as m = 3, and the frequency is chosen as 4000 Hz. It is found that the wavenumbers are not symmetric about the origin because of the uniform mean flow. 400 Right Left
Imag(k) [1/m]
200
lining-borne mode
air-borne mode
0
-200 air-borne mode
-400 -100
-50
lining-borne mode
0 Real(k) [1/m]
50
100
Fig. 6. Axial wavenumbers at 4000 Hz with M = 0.2 and m = 3; right- and left-travelling modes are marked with blue asterisk and red cross marks respectively; air-borne and lining-borne modes are marked respectively.
16
(a) air-borne modes
(b) Lining-borne modes
17
Fig. 7. Mode shapes of first 3 eigenmodes at 4000 Hz with M = 0.2 and m = 3; (a) right- and lefttravelling air-borne (3,0)a, (3,1)a and (3,2)a modes; (b) right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes. In addition, in both right-travelling modes and left-travelling modes, the eigenvalues can be divided into air-borne modes and lining-borne modes. The mode shapes for pressure field of first 3 air-borne modes are shown in Fig. 7(a). It can be seen that the mode shapes of air-borne modes for both righttravelling and left-travelling waves are nearly the same, which mainly propagate in the air domain, while the pressure in the porous material domain is relatively small. In addition, with the increase of radial order, the number of cross-zero points in both air-domain and porous material domain increases. The mode shapes for pressure field of first 3 lining-borne modes are shown in Fig. 7(b). In contrast to the air-borne modes, the mode shapes of lining-borne modes are different for right-travelling and lefttravelling waves, when there is uniform mean flow. For the right-travelling modes, in other words, the direction of the wave propagation is the same as flow direction, the wave mainly propagates in lining domain for lining-borne modes. For the left-travelling modes, which travels in negative flow direction, some of the lining-borne modes propagate only in lining domain (e.g. (3,0)l mode), while some in both lining domain and air domain (e.g. (3,1)l mode). In other words, the main characteristic for the liningborne modes is that they can all propagate in porous lining domain, while some of them also propagate in the air domain but some do not. Therefore, the modes for left-travelling wave are more influenced by the uniform mean flow, which will be further described in the next section. As the mode properties may change along with frequency, the mode classification is conducted according to the calculation result at 4000 Hz with Mach number M = 0.2. The influence of frequency on the eigenmodes is investigated by the kinetic energy ratio RKE in two domains, which is inspired by Ref. [38], and defined as RKE
Ekl , Eka
(20)
where Ekl and Eka are averaged kinetic energy in lining domain and air domain, respectively. For the airborne mode, RKE is generally much larger than 1, while for lining-borne mode, RKE is around 1 or much smaller than 1. In the lining domain, there is no mean flow, so the kinetic energy can be simply calculated by the kinetic energy with acoustic particle velocity, as
Ekl
1 2 Re c v dS 4 lining domian
Sl
18
,
(21)
where v is the acoustic particle velocity and Sl is the total area of lining domain. In the air domain, uniform mean flow exists, so there is no strict criterion to separate kinetic energy from total acoustic energy. Moreover, it is still a subject of discussion that which part of the field is to be called acoustic energy with the existence of flow field [39]. Therefore, the analog of definition in Eq. (21) is used to represent the kinetic energy in air domain Eka , which is defined as 1 2 0 v dS 4 Ekl air domian , Sa
(22)
where v is the acoustic particle velocity and Sa is the total area of air domain. The evolution of kinetic energy ratio with frequency for different modes is displayed in Fig. 8(a)-(d). It is found that with the change of frequency, higher order right- and left-travelling air-borne modes, e.g. (3,1)a mode and (3,2)a mode, generally keep their properties as air-borne modes. For both right- and lefttravelling air-borne (3,0)a mode, it performs like a lining mode in the low-frequency range (with RKE around 1), and gradually evolves to the air mode in the high-frequency range. The evolution of liningborne modes is different from air-borne modes. For both right- and left-travelling lining-borne (3,0)l mode, it keeps as lining-borne mode with the change of frequency. On the contrary, for higher order right- and left-travelling lining-borne modes, e.g. (3,1)l mode and (3,2)l mode, they show the properties of air-borne modes in the low-frequency range, and evolve to lining-borne modes with the increase of frequency.
19
20
Fig. 8. Evolution of kinetic energy ratio RKE with frequency for different modes; (a) right-travelling airborne (3,0)a, (3,1)a and (3,2)a modes; (b) left-travelling air-borne mode (3,0)a, (3,1)a and (3,2)a modes; (c) right-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes; (d) left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes. 4.2
Effect of Mach number on eigenmodes The effect of Mach number on the axial wavenumbers and mode shapes are described in this section.
The range of Mach number is set as [0, 0.5], and frequency is chosen as 4000Hz. The wavenumber evolution with the increase of Mach number is demonstrated in Fig. 9. It is found that, the effect of Mach number on air-borne modes and lining-borne modes are different. The change of Mach number will make greater changes of wavenumbers for air-born modes, compared with its effect on lining-borne modes. For the left-travelling air-borne modes, the increase of Mach number will mainly increase the real part of the wavenumber. It can be easily understood that increase of Mach number will effectively decrease the speed of sound for the waves travelling in negative flow direction, which in turn increases the real part of the wavenumber. Compared with real part, the effect of Mach number on imaginary part is small. For the right-travelling air-borne modes, the real part of some lower order modes ((3,0)a mode in this calculation example) will decrease with the increase of Mach number, which can be explained by the effectively increased speed of sound; while the real part of higher order modes (other modes in this calculation example) will increase with the increase of Mach number, which is due to the effect of flow convection. Generally speaking, the increase of Mach number will shift the wavenumbers of air-borne modes to the left on the complex plane of wavenumbers. 21
Fig. 9. Effect of Mach number on axial wavenumbers for right-travelling (solid line) and left-travelling (dash line) modes with M = 0~0.5; circular markers for M = 0 and triangular markers for M = 0.5. For right-travelling lining-borne modes, the change of Mach number has little effect on the wavenumber. For some particular left-travelling lining-borne mode (e.g. (3,2)l in this calculation example), the behaviour of the wavenumber with the change of Mach number is more like an air-borne mode. It is further supported by the its mode shape, which will be described in the following. The aforementioned result is summarized in Table 1 for reference. Table 1 Axial wavenumbers of the calculation example at 4000Hz. M=0, Right
M=0, Left
M=0.5, Right
M=0.5, Left
(3,0)a
54.83-2.25i
-54.83+2.25i
34.09-3.74i
-120.74+3.70i
(3,0)l
78.54-10.10i
-78.54+10.10i
78.45-10.26i
-78.74+9.84i
(3,1)a
2.18-38.95i
-2.18+38.95i
-19.95-15.41i
-46.67+29.97i
(3,1)l
69.31-10.75i
-69.31+10.75i
68.34-12.70i
-72.93+9.16i
(3,2)a
0.73-89.20i
-0.73+89.20i
-40.44-85.32i
-45.63+96.94i
(3,2)l
42.59-16.29i
-42.59+16.29i
41.98-20.83i
-57.93+15.93i
(3,3)a
0.42-127.91i
-0.42+127.91i
-43.30-133.00i
-44.30+144.35i
(3,3)l
11.80-60.08i
-11.80+60.08i
15.92-63.69i
-17.66+59.81i
22
(a) M = 0
(b) M = 0.5 Fig. 10. Effect of Mach number on mode shapes for right- and left-travelling air-borne (3,0)a, (3,1)a and (3,2)a modes at 4000 Hz; (a) M = 0; (b) M = 0.5. 23
The mode shapes of air-borne modes are shown in Fig. 10(a) and (b), respectively for M = 0 and M = 0.5. It is found that, for M = 0, in other words, there is no mean flow in the duct, the mode shapes for right-travelling and left-travelling waves are exactly the same. When there is uniform mean flow with M = 0.5, the mode shapes are changed a little bit from the case with M = 0. However, for both left-travelling and right-travelling waves, the effect of Mach number on the mode shapes of air-borne modes are small. The mode shapes of lining-borne modes are shown in Fig. 11(a) and (b), respectively for M = 0 and M = 0.5. It is found that for right-travelling modes, the increased Mach number slightly changes the mode shape. For left-travelling modes, the mode shapes are strongly affected by the change of Mach number. In the calculation example, the (3,0)l mode mainly propagates in porous lining domain with M = 0, while it can propagate in both porous lining domain and air domain when the Mach number increases to 0.5. For the (3,2)l mode, it mainly propagates in porous lining domain with M = 0, while it changes to performs like an air-borne mode M = 0.5, which is in accordance with the trend of corresponding wavenumber.
(a) M = 0
24
(b) M = 0.5 Fig. 11. Effect of Mach number on mode shapes for right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes at 4000 Hz; (a) M = 0; (b) M = 0.5. It is summarized that, for air-borne modes, Mach number will strongly affect their wavenumbers, but its effect on the mode shapes is small. On the contrary, for lining-borne modes, Mach number has small effect on wavenumbers, but its effect on mode shapes is strong, even the mode type can be changed from a lining-borne mode to an air-borne mode for some particular left-travelling mode. 4.3
Effect of Mach number on transmission loss The transmission loss (TL) of a lined duct with mean flow is calculated with the mode matching
method [1], which is described in detail in Appendix A, to further validate the calculated eigenmodes and corresponding axial wavenumbers. In addition, the effect of Mach number on the TL is also analysed. In this calculation example, the plane wave incidence condition is applied. The procedure of the mode matching scheme is briefly described here for completeness. The axialsectional view of calculation geometry is shown in Fig. 5 (a). Regions I and III are air ducts with radius r1 and sound wave comes from the left. Region II is porous lined duct with radius r2, while the air domain has the same radius as Region I and III. The length of the lined part is L, and the length of the air duct is L0, which is chosen to be the same as L in this calculation example. The waves inside each region (I, II and III) are expressed as the superposition of eigenmodes of the right-travelling and left-travelling waves with the unknown mode amplitudes to be solved. Applying the boundary condition on the two interfaces 25
Γ1 and Γ2 (as shown in Fig. 5 (a)), the equation for the unknow mode amplitudes can be obtained, and the TL can be then obtained through calculated mode amplitudes. For Region II, the eigenmodes and wavenumbers are obtained with the WFE method. For Regions I and III, analytical solutions for the eigenmode and axial wavenumber exist, described as Eqs. (18)-(19). The axial wavenumbers and corresponding eigenmodes are sorted out by the absolute value of the imaginary part of wavenumbers in ascending order, ensuring that the least attenuated modes are included in the mode matching scheme. The parameters of the lined duct geometry and lining porous materials as well as perforated panel are listed in Table 2. Table 2 Geometry and material parameters. Lined duct geometry
Perforated panel
Porous material
Length L = L0 [mm]
300
Thickness th [mm]
1
Radius of air duct r1 [mm]
100
Hole diameter d [mm]
1
Radius of lined duct r2 [mm]
100
Porosity σ
5%
Flow resistivity Rf [Pa·s/m2]
5000
The TL of the lined duct with M = 0.2 is shown in Fig. 12 (solid line), together with the result from full FE simulation (circular points). It is found that, the TL calculated with eigenmodes matches well with that from full FE simulation. In this case, the eigenmodes and eigenvalues of the porous lined duct with uniform mean flow are validated.
12
TL [dB]
10 8 6 Mode matching Full FE
4 2
0
500
1000 Frequency [Hz]
1500
2000
Fig. 12. Validation of transmission loss (TL) calculated by mode matching with full FE simulation for M = 0.2.
26
Fig. 13. Effect of Mach number on transmission loss (TL) of lined duct with three Mach numbers: M = 0, M = -0.2 and M = 0.2. The effect of Mach numbers on TL is investigated and shown in Fig. 13. It is found that, when the Mach number is negative, the TL is increased for both low-frequency range and high-frequency range. However, the peak value for TL is decreased with negative Mach number. In addition, the peak frequency is nearly not shifted. When the Mach number is positive, TL is generally smaller than the case of M = 0. In other words, TL will generally increase with up-flow (M = -0.2), except for frequencies around the peak; while it will generally decrease with down-flow (M = 0.2). In both cases, the peak frequency does not show significant change. The effect of flow resistivity Rf on the TL is investigated, and the result is shown in Fig. 14 for the calculated example with M = 0.2. It is found that with the increase of Rf, the peak value of TL decreases and the high frequency TL increases. In addition, the peak frequency shifts to higher frequency range with the increase of Rf. These characteristics are in accordance with the case of no mean flow [40].
27
Fig. 14. Effect of flow resistivity Rf (5000, 10000 and 20000 Pa·s/m2) on transmission loss (TL) of lined duct with M=0.2. 5
Conclusions In this paper, the effect of uniform mean flow on the eigenmodes of lined ducts is analyzed with a
combined wave and finite element (WFE) method. The duct is lined with bulk-reactive porous materials, which is separated from the flow channel by a perforated panel. The proposed method is validated by the analytical solution and the convergence rate of the method is discussed. It is found that the method is applicable for Mach number M at least up to 0.8. With properly chosen of segment length , the WFE method can reach the convergence rate of Se4 for eigenvalues with M ≤ 0.8, where Se is the mesh size. Nevertheless, smaller mesh size is needed for higher order modes to achieve the same accuracy as lower order modes. With porous lining and perforated panel, the eigenmodes can be divided into air-borne modes and lining-borne modes for both right-travelling and left-travelling waves. The air-borne modes propagate mostly in the air domain, while the lining-borne modes can propagate in both air domain and porous domain. The effect of Mach number on the axial wavenumbers and mode shapes are analysed. It is found that, Mach number will strongly affect the wavenumbers of air-borne modes. The increase of Mach number will shift the wavenumbers of air-borne mode to the left on the complex plane of wavenumbers. However, the effect of Mach number on mode shapes of air-borne modes is small. On the contrary, for the liningborne modes, Mach number has small effect on wavenumbers, but its effect on mode shapes is strong,
28
even the mode type can be changed from a lining-borne mode to an air-borne mode for some lefttravelling modes. The transmission loss (TL) of a typical porous lined duct with three typical Mach numbers M = 0, 0.2 and -0.2 is calculated with the eigenmodes by the mode matching method. The calculated TL matches well with the result from full finite element simulation, which further validates the calculated eigenmodes. It is found that up-flow (M = -0.2) will generally increase TL of lined duct, compared with the case of no flow, except for frequencies around the peak; while down-flow (M = 0.2) will generally decrease TL. In both cases of up-flow and down-flow, the peak frequency does not show significant change. Acknowledgment Part of the work was accomplished when the first author was the Post-Doctoral Fellow in The University of Hong Kong. This project is supported by National Natural Science Foundation of China (Grant No. 51775467).
AUTHOR DECLARATION We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
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[22] J.M. Renno, B.R. Mace, On the forced response of waveguides using the wave and finite element method, J. Sound Vib., 329 (2010) 5474-5488. [23] T. Kohrs, B.A.T. Petersson, Wave beaming and wave propagation in light weight plates with trusslike cores, J. Sound Vib., 321 (2009) 137-165. [24] E. Manconi, B.R. Mace, Wave characterization of cylindrical and curved panels using a finite element method, J. Acoust. Soc. Am., 125 (2009) 154-163. [25] J.F. Allard, Y. Champoux, New empirical equations for sound propagation in rigid frame fibrous materials, J. Acoust. Soc. Am., 91 (1992) 3346-3353. [26] M.E. Delany, E.N. Bazley, Acoustical properties of fibrous absorbent materials, Appl. Acoust., 3 (1970) 105-116. [27] D.L. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluidsaturated porous media, J. Fluid Mech., 176 (1987) 379-402. [28] Y. Champoux, J.F. Allard, Dynamic tortuosity and bulk modulus in air-saturated porous media, J. Appl. Phys., 70 (1991) 1975-1979. [29] Y. Miki, Acoustical properties of porous materials-modifications of Delany-Bazley models, J. Acoust. Soc. Jpn. (E), 11 (1990) 19-24. [30] R. Kirby, On the modification of Delany and Bazley fomulae, Appl. Acoust., 86 (2014) 47-49. [31] S. Allam, M. Abom, A new type of muffler based on microperforated tubes, J. Vib. Acoust.-Trans. ASME, 133 (2011). [32] R. Kirby, A. Cummings, The impedance of perforated plates subjected to grazing gas flow and backed by porous media, J. Sound Vib., 217 (1998) 619-636. [33] I. Lee, A. Selamet, N. T Huff, Acoustic impedance of perforations in contact with fibrous material, J. Acoust. Soc. Am., 119 (2006) 2785-2797. [34] U. Ingard, Perforated facing and sound absorption, J. Acoust. Soc. Am., 26 (1954) 151-154. [35] M.L. Munjal, Acoustics of ducts and mufflers, 2 ed., John Wiley & Sons, 2014. [36] W. Bi, V. Pagneux, D. Lafarge, Y. Auregan, An improved multimodal method for sound propagation in nonuniform lined ducts, J. Acoust. Soc. Am., 122 (2007) 280-290. [37] E.J. Brambley, A.M.J. Davis, N. Peake, Eigenmodes of lined flow ducts with rigid splices, J. Fluid Mech., 690 (2012) 399-425. [38] B. Takiuti, E. Manconi, M. Brennan, V.L. Junior, Wave transmission from asymmetrical changes of cross-sectional area in a beam, in: Thirteenth International Conference on Recent Advances in
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Structural Dynamics (RASD), Journal of Physics: Conference Series, 15–17 April 2019, Valpre, Lyon, France, 2019, pp. 012056. [39] S.W. Rienstra, A. Hirschberg, An introduction to acoustics, Eindhoven University of Technology, 2018. [40] I.L. Vér, L.L. Beranek, Noise and vibration control engineering: principles and applications, 2 ed., Wiley Online Library, 2006.
32
Appendix A.
Mode matching sheme [1] The calculation geometry setting is shown in Fig. 5 (a). The waves inside each region (I, II and III)
are expressed as the superposition of eigenmodes of the right-travelling and left-travelling waves with the unknown mode amplitudes to be solved. For example, the air domain pressure in Region II is expressed as Nm
p a ,II r , z AnII, pII,a ,n r e n 1
izk zII,n,
AnII, pII,a ,n r e
izk zII,n,
,0 r r , 1
(A-1)
where, the superscript a denotes the air part; the superscript II denotes region II; symbols + and – denote the right-travelling and left-travelling waves, respectively, pII,a , n and pII,a , n are air domain pressure parts of the corresponding n-th eigenmodes ; k zII,,n and k zII,,n are the axial wavenumbers; AnII, and AnII, are the unknown mode amplitudes , Nm is the number of eigenmodes used. Changing the above equation into the matrix form gives
p a ,II r , z pII,a r EII, z AII, pII,a r EII, z AII, ,0 r r1 ,
(A-2)
II,a / r II,a / r , II,a / r , , II,a / r p ,2 p , Nm p ,1 p II , / II, / II , / II , / izk z diag eizk1 ,eizk2 , ,e Nm . E A II, / AII, / , AII, / , , AII, / T 2 Nm 1
(A-3)
where,
Applying the continuity of pressure and normal component of particle displacement on air part of the two interfaces Γ1 and Γ2 (0
(A-4)
A III, A III, X2 E X 2 II, Y2 EY 2 II, , A A
(A-5)
with
PaI, X1 U I,a 0
PaI, PaII, I, U II, a , Y1 U a 0 WpII,
PaIII, PaII, U aII, , X2 U III, a 0 WpII, 33
PaIII, PaII, III, U II, a , Y2 U a 0 WpII,
PaII, U aII, , WpII,
(A-6)
EIII, EX 2 0
EIII, 0 , E z Y2 EII, 0
0 . EII,
(A-7)
Al,s (l = I, II, III and s = +,–) is a column vector with the length of Nm, containing the right-travelling or left-travelling mode amplitudes. El,t (l = I, II, III and s = +,–) is a diagonal matrix, which is defined as
El , s diag e
izk zl ,,1s
,e
izk zl ,,2s
,e
izk zl ,,3s
, ,e
izk zl ,,sN
m
. The components in matrix X , Y and X , Y are scattering 1
1
2
2
matrix calculated by boundary integration on the interface, which is defined as follows. On the left boundary Γ1, † r1 PaI, / ψ pa pI,a / rdr 0 † r II, / 1 ψ p a pII,a / rdr Pa 0 † r1 I, / I, / U a ψ uza uza rdr , 0 † r1 II, / U II, / ψ 0 uza uza rdr a † II , / r2 ψ u p uIIp, / rdr Wp z z r1
(A-8)
with ψ pa pI,a , ψua uII,a , ψu p uII,p , and the superscript † referring to take transpose conjugate of a z
z
z
z
vector. On the right boundary Γ2, † r1 III, / Pa ψ pa pIII,a / rdr 0 † r1 II, / ψ p a pII,a / rdr Pa 0 † r III, / 1 / ψ u a uIII, rdr , a U a z z 0 † r1 II, / U II, / ψ 0 uza uza rdr a † II , / r2 ψ u p uIIp, / rdr Wp z z r1
(A-9)
with ψ pa pIII,a , ψua uII,a , ψu p uII,p . z
z
z
z
In the present calculation example, a plane wave incidence with unit amplitude is assumed, so that AI,+ = [1,0,0…,0]T. On the outlet interface, a non-reflecting boundary condition is applied with AIII,– = 34
[0,0,0…,0]T. The unknown mode amplitudes are obtained by solving Eqs. (A-4) and (A-5). The TL of the lined duct is then found as
TL 20log10 A1III, . B.
(A-10)
Matrix in the eigenvalue equation In Eq. (12), the new partitioned dynamic stiffness matrixs DLL, DLR, DRL and DRR are expressed as D LL D LL D LI DII1D IL 1 D LR D LR D LI D II D IR . 1 D D D D D RL RL RI II IL 1 D RR D RR D RI D II D IR
(B-1)
In Eq. (13), the elements in the transfer matrix T are expressed as
DLR1 DLL T 1 DRL DRR DLR DLL
35
DLR1 . DRR DLR1
(B-2)
Figure captions Fig. 1.
Geometry of the porous lined duct with arbitrary cross-section containing a uniform mean flow U0; (a) axial-sectional view with axial direction set as z axis; (b) cross-sectional view with 2D coordinate system; (c) 3D view for a short segment with length .
Fig. 2.
Effect of segment length Δ on wavenumbers (relative error compared with analytical value) for right-travelling (1,1), (3,3) and (5,5) modes; (a) M = 0; (b) M = 0.5; (c) M = 0.8.
Fig. 3.
Convergence of wavenumbers (relative error compared with analytical value) with mesh size Se for right-travelling (1,1), (3,3) and (5,5) modes, together with the trend of Se4 provided for reference; (a) M = 0; (b) M = 0.5; (c) M = 0.8.
Fig. 4.
Convergence of wavenumber with mesh size Se at three different frequencies, shown as relative error compared to analytical value, for the (3,3) right-travelling mode, together with the trend of Se4 provided for reference
Fig. 5.
Geometry of the porous lined duct with circular cross-section; (a) axial sectional view; (b) cross-sectional view.
Fig. 6.
Axial wavenumbers at 4000 Hz with M = 0.2 and m = 3; right- and left-travelling modes are marked with blue asterisk and red cross marks respectively; air-borne and lining-borne modes are marked respectively.
Fig. 7.
Mode shapes of first 3 eigenmodes at 4000 Hz with M = 0.2 and m = 3; (a) right- and lefttravelling air-borne (3,0)a, (3,1)a and (3,2)a modes; (b) right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes.
Fig. 8.
Evolution of kinetic energy ratio RKE with frequency for different modes; (a) right-travelling air-borne (3,0)a, (3,1)a and (3,2)a modes; (b) left-travelling air-borne mode (3,0)a, (3,1)a and (3,2)a modes; (c) right-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes; (d) left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes.
Fig. 9.
Effect of Mach number on axial wavenumbers for right-travelling (solid line) and left-travelling (dash line) modes with M = 0~0.5; circular markers for M = 0 and triangular markers for M = 0.5.
Fig. 10.
Effect of Mach number on mode shapes for right- and left-travelling air-borne (3,0)a, (3,1)a and (3,2)a modes at 4000 Hz; (a) M = 0; (b) M = 0.5.
Fig. 11.
Effect of Mach number on mode shapes for right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes at 4000 Hz; (a) M = 0; (b) M = 0.5.
36
Fig. 12.
Validation of transmission loss (TL) calculated by mode matching with full FE simulation for M = 0.2.
Fig. 13.
Effect of Mach number on transmission loss (TL) of lined duct with three Mach numbers: M = 0, M = -0.2 and M = 0.2.
Fig. 14.
Effect of flow resistivity Rf (5000, 10000 and 20000 Pa·s/m2) on transmission loss (TL) of lined duct with M = 0.2.
Table captions Table 1
Axial wavenumbers of the calculation example at 4000Hz.
Table 2
Geometry and material parameters. Table 1 Axial wavenumbers of the calculation example at 4000Hz. M=0, Right
M=0, Left
M=0.5, Right
M=0.5, Left
(3,0)a
54.83-2.25i
-54.83+2.25i
34.09-3.74i
-120.74+3.70i
(3,0)l
78.54-10.10i
-78.54+10.10i
78.45-10.26i
-78.74+9.84i
(3,1)a
2.18-38.95i
-2.18+38.95i
-19.95-15.41i
-46.67+29.97i
(3,1)l
69.31-10.75i
-69.31+10.75i
68.34-12.70i
-72.93+9.16i
(3,2)a
0.73-89.20i
-0.73+89.20i
-40.44-85.32i
-45.63+96.94i
(3,2)l
42.59-16.29i
-42.59+16.29i
41.98-20.83i
-57.93+15.93i
(3,3)a
0.42-127.91i
-0.42+127.91i
-43.30-133.00i
-44.30+144.35i
(3,3)l
11.80-60.08i
-11.80+60.08i
15.92-63.69i
-17.66+59.81i
Table 2 Geometry and material parameters. Lined duct geometry
Perforated panel
Porous material
Length L = L0 [mm]
300
Thickness th [mm]
1
Radius of air duct r1 [mm]
100
Hole diameter d [mm]
1
Radius of lined duct r2 [mm]
100
Porosity σ
5%
37
Flow resistivity Rf [Pa·s/m2]
5000
Graphical_abstract
38
On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow Changyong Jiang , Chunqi Wang PII: DOI: Reference:
S0020-7403(19)31710-2 https://doi.org/10.1016/j.ijmecsci.2019.105300 MS 105300
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
15 May 2019 11 October 2019 2 November 2019
Please cite this article as: Changyong Jiang , Chunqi Wang , On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105300
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Highlights Eigenmode analysis of lined flow duct with arbitrary cross section Two kinds of eigenmodes are observed and characterized Mode properties change with frequency and Mach number Mach number affects wavenumber of air-borne mode but mode shape of lining-borne mode
1
On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow Changyong Jianga,b,c, and Chunqi Wangb,c,* a
College of Engineering Physics, Shenzhen Technology University, Shenzhen 518118, China b
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China
c
Laboratory for Aerodynamics and Acoustics, HKU-Zhejiang Institute of Research and Innovation, Lin An 311305, Zhejiang, China *Corresponding author, E-mail: [email protected] Phone: 852-28592621; Fax: 852-28585415
Abstract: The present work concerns the numerical eigenmode analysis of acoustically lined ducts of any constant cross-sectional shape in the presence of mean flow. The duct is lined with bulk-reactive porous material, which is separated from the flow channel by a perforated panel. A combined wave and finite element (WFE) method is introduced to investigate the effects of uniform mean flow on the eigenmodes and eigenvalues of the lined ducts. The proposed method is validated against analytical solutions for a simple case. A convergence rate of Se4 for eigenvalues can be reached for Mach number up to 0.8, with Se being the mesh size. In this study, the eigenmodes of a lined duct are divided into air-borne modes and liningborne modes. The air-borne modes propagate mostly in the air domain, while the lining-borne modes can propagate in both air domain and porous domain. The effect of Mach number on the axial wavenumbers and mode shapes are analysed. It is found that, for air-borne modes, Mach number will strongly affect their wavenumbers, while its effect on the mode shape is marginal. On the contrary, for the lining-borne modes, Mach number has small effect on the wavenumber, but its effect on the mode shape is significant. Specifically, even the mode properties can be changed from a lining-borne mode to an air-borne mode for some left-travelling lining-borne mode. The transmission loss (TL) of a lined duct with three typical Mach numbers is calculated by the mode matching method. It is found that up-flow will generally increase TL of lined duct, compared with the case of no flow; while down-flow will generally decrease TL. In both situations, the peak frequency does not show significant change. Keywords: eigenmode analysis; mean flow; duct lining; eigenvalue problem; finite element method
2
1
Introduction Acoustically lined ducts are commonly used in various engineering applications for noise control
purpose, like heating, ventilation, and air conditioning (HVAC) systems, automobile cars, and aviation industry. In a previous research, a poroelastic lined duct is analyzed based on the calculated crosssectional eigenmodes [1], but the effect of flow was not considered. In real applications like HVAC systems, the air passage generally contains flow, which may affect the lining properties seriously. The present work concerns the numerical eigenmode analysis of acoustically lined ducts of any constant cross-sectional shape in the presence of mean flow. The effect of uniform mean flow on the eigenmodes is focused. In the literature, various methods have been developed to calculate the eigenmodes of acoustic duct linings with mean flow. An extensive review on this subject can be found by Kirby in 2009 [2]. When the cross-section has a simple, regular shape (rectangular, circular or annular), an analytical eigenvalue problem for the duct lining can be formulated [3, 4]. Generally, a robust root-finding algorithm is needed to find solutions of the eigenvalue equation, which is not a trivial task [5]. Recently, many researchers have made new investigations into this topic. Jiang et al. proposed an analytical method to model the lined duct with azimuthally non-uniform impedance [6]. In their modelling, Fourier series expansion was incorporated into the Wiener-Hopf method and uniform mean flow in the duct was considered. Qiu et al. formulated the analytical eigenvalue problem of lined annular duct with uniform mean flow [7]. In their research, the Cremer concept [8] was further investigated to optimize the surface impedance, with which maximum noise reduction for the eigenmode could be achieved. Bravo and Maury investigated sound attenuation of non-locally lining structures by anisotropic fibrous materials in contact with uniform mean flow [9]. The least attenuated mode of the lining system was obtained with a simulated annealing search method. In addition to the aforementioned frequency domain method, time domain method to obtain eigenmodes has also been proposed [10]. For complex cross-sectional shapes, an analytical formulation is usually impossible. It is an attractive alternative to solve the eigenvalue relationship numerically to avoid root-finding, like the point-matching method [11], Chebyshev collocation method [12] and so on, but a regular cross-sectional geometry is a prerequisite. Recently, a semi-analytical formulation was developed to investigate the acoustic eigenproblems of three-dimensional elliptical cylindrical cavities with multiple elliptical cylinders [13]. Having said these, a discrete formulation for the cross-sectional eigenmodes are more preferred with arbitrary geometry, e.g. the finite element (FE) formulation and mesh-free methods. A FE approach is more versatile for any type of cross-sectional geometry. In this regard, Son et al. provided the FE formulation of eigenvalue problem for poroelastic lined duct [14]. In their study, the influence of flow
3
and perforated panel were ignored. Astley and Cummings established the eigenvalue problem with FE formulation for a duct lined with bulk-reactive porous material, while considering the uniform mean flow [15]. Kirby took the influence of perforated panel into consideration, as well as the uniform mean flow [16]. Recently, Denia et al. obtained the eigenvalues and associated eigenmodes for the silencer’s twodimensional (2D) cross section with FE formulation [17]. Effect of uniform mean flow and transverse temperature gradient was included in the model. A point collocation technique was then used to predict transmission loss (TL), which matched favorably well with a full three-dimensional (3D) FE simulation, with reduced computational effort. Sanchez-Orgaz et al. obtained the eigenmode of 2D cross-section of a lined duct through FE formulation, with the effect of perforated panel considered [18]. The granular material was used as sound absorption material, but the effect of flow was not considered. TL was also obtained through mode matching method. Fang and Liu investigated eigenmodes of the silencer crosssection through a semi-weak-form mesh-free method [19]. In their method, shape functions were created with the radial basis function point interpolation method. Galerkin method was adopted to obtain the weak formulation of transversal governing equations, and the Gauss integration scheme was used to compute the system matrices. However, a discrete formulation of the eigenvalue problem by direct FE or mesh-free method requires specific prior knowledge of the application procedure, which is beyond the capability of many engineers. It is therefore desirable to develop an alternative approach that is more efficient to apply. This study seeks to investigate the effect of uniform mean flow on the eigenmode properties of an acoustically lined ducts of any constant cross-sectional shape with a wave and finite element (WFE) method. With this method, the effect of bulk-reactive porous material and perforated panel can be easily taken into account. In addition, the method can be applied to acoustic duct linings with arbitrary crosssectional shape, as long as it is uniform along the axial direction. The general idea of the WFE method was first proposed in [20] to analyze the one-dimensional wave propagation problems. The major advantage of such method is that it formulates the eigenvalue problem based on general FE matrices, which can be obtained conveniently from commercially available FE softwares. Hence, the modelling procedure of even complicated problems becomes relatively easy to handle which is of considerable practical interest. The WFE method has been successfully applied to investigate wave propagation problems in beams [21, 22] and plates [23, 24]. In this paper, the eigenmodes of porous lined duct with uniform mean flow is investigated and analyzed. Different types of modes are identified base on mode properties, and mode evolution with frequency and Mach number is investigated. The rest of the paper is structured as follows. In Sec. 2, the modelling and eigenvalue problem formulation procedure are described. In Sec. 3, the proposed approach is validated with analytical solutions. In Sec. 4, the effect of Mach number on the eigenmodes of the lined duct is presented. Conclusions are drawn in Sec. 5.
4
2 2.1
Methodology Problem description In the present study, a porous material lined duct with uniform mean flow in axial direction is
considered, whose cross-section can be arbitrary in shape. Between the air domain and porous lining domain, a perforated panel is used. The axial sectional, cross-sectional and three dimensional (3D) views are shown in Fig. 1(a)-(c), respectively. The acoustic wave in the air domain of the lined duct is described by the convected wave equation, 2 p a
1 D2 pa 0, c02 Dt 2
(1)
where pa is the acoustic pressure for air domain, c0 is the speed of sound in stationary air, D2 D U 0 , and U0 is the mean flow velocity. The calculation is conducted in 2 Dt z Dt t 2
2
frequency domain, where the time dependence of eit is assumed, ω = 2f is angular frequency, f is frequency with the unit of Hz and i 1 . Therefore, Eq. (1) can be reformed as,
2 p a 1 M 2
zp 2
2
a
2ik0 M
p a k02 p a 0 , z
(2)
where 2 is the Laplace operator in the cross-section, M = U0/c0 is the Mach number, k0 = /c0 is the wavenumber of sound in air. The particle displacement in air domain ua can be calculated with Euler equation
0
Du a p a . Dt
(3)
with ρ0 being the air density. For the porous lining domain, the porous material is modelled with the semi-empirical model proposed by Allard and Champoux [25]. In the literature, Delany and Bazely’s empirical model [26] and Johnson-Champoux-Allard’s (JCA) physical model [27, 28] are widely used. However, both of them have some restrictions. Delany model is valid in a limited frequency range, which may cause unphysical results in low-frequency range. Although Miki [29] has made modifications to Delany’s model, problem in lowfrequency is not completely solved [30]. JCA model more accurate with physical bases, which can also be adopted here as in the previous literature [1], but it is more complicated with several parameters which is not easy to determine, like viscous and thermal characteristic lengths. Therefore, the semi-empirical model [25] is adopted in this study, which is simple to implement with one parameter while keeping meaningful results in low-frequency range. With this model, the porous material is treated as an equivalent fluid with its equivalent density ρc and elastic modulus Kc expressed as 5
2 1 1.2 0.0364 0 f i0.1144 0 f Rf Rf c 2 0 f f i29.64+ 2.82 i24.9 0 Rf Rf K c 101320 2 0 f f i21.17+ 2.82 i24.9 0 Rf Rf
1
1
,
(4)
where Rf is the flow resistivity. The equivalent speed of sound cc, characteristic impedance zc and wavenumber kc of porous materials are calculated as: cc K c c . zc c cc k z c c
(5)
The acoustic field in the porous lining domain is governed by the Helmholtz equation with the equivalent fluid model,
2 p p kc2 p p 0 ,
(6)
where pp is the acoustic pressure in porous lining domain. The particle displacement in the porous domain up is then obtained as
up
1
c
2
p p .
(7)
The perforated panel between the air domain and porous lining domain is modelled with the model proposed by Allam and Abom [31]. In this model, the perforated panel is treated as an impedance boundary with impedance zpp z0 rpp xpp i , where z0 = ρ0c0 is the characteristic impedance of air, rpp and xpp are resistance and reactance, respectively, determined as
i i
2 J1 i t h r Re 1 pp c i J0 0 2 J1 i t xpp Im h 1 c0 i J 0
1
i i
1
K pp M uˆh 2 Rs c c 0 0 0 uˆh pp F 1 c0 c0
1
.
(8)
where Re and Im mean taking real part and imaginary part of a complex variable respectively, J0 and J1 mean the zero-th and first order of Bessel function, respectively, σ is perforated ratio of the panel, th is the thickness of the perforated panel, d 4 is a dimensionless variable, relating to the diameter of the 6
holes in the panel d, and kinetic viscosity of air , Rs
1 20 is the surface resistance, η is the 2
dynamic viscosity of air, pp = 8d/3 is the end correction of the hole, uˆh is the averaged particle velocity
3 1
in the hole, Kpp 0.15 0.0125 std represents the effect of flow on resistance, F 1 12.6M represents the effect of flow on reactance.
Note that Eq. (8) is used for air-perforated panel-air system, which is a little bit different from the case described here. Actually, when the porous material is in close contact with the perforated panel, the impedance of the perforated panel can be much more complicated, as investigated by Kirby and Cummings for the case with mean flow [32] and Lee et al. for the case without mean flow [33]. In Kirby and Cummings research [32], it was concluded that the proposed semi-empirical scheme yielded good result for resistance and at least adequate result for the mass end correction. In practice, it is difficult to ensure a perfect close contact between the perforated panel and the porous material. Hence, a small air gap greater than the hole diameter is assumed between the perforated panel and the porous material, as adopted and experimentally validated by Bravo and Maury [9]. In this case, the effect of porous material on the panel impedance can be neglected. If air gap is smaller than the hole diameter, impedance of the perforated panel should be modified [32-34]. In the present study, assumption by Bravo and Maury [9] is used, so the influence of porous material on perforated panel is not taken into consideration. At the perforated panel, which is air-porous domain interface, referring to Fig. 1, the following compatibility conditions are applied,
u a n u p n , a p p p zpp vn
(9)
where n is the normal vector of the interface, vn is the normal component of particle velocity in porous lining domain.
7
Porous material Perforated panel L I R
Air
U0
z axis
(a) Axial-sectional view
y
z axis Air 0
x
Porous material
(b) Cross-sectional view
(c) 3D view
Fig. 1. Geometry of the porous lined duct with arbitrary cross-section containing a uniform mean flow U0; (a) axial-sectional view with axial direction set as z axis; (b) cross-sectional view with 2D coordinate system; (c) 3D view for a short segment with length . 2.2
Eigenvalue problem formulation The eigenvalue problem for the cross-section of the porous lined duct is formulated using WFE
method [1, 20]. In this formulation, the acoustic field on the cross-section forms the eigenmode, and the corresponding axial wavenumber forms the eigenvalue. Therefore, the solved eigenmode is for the 2D cross-section, and the eigenvalue depends on the frequency. The major steps of the procedure are presented below for completeness. A short segment of the lined duct with axial length , which is marked in Fig. 1(a) and depicted in 3D form in Fig. 1(c), is modelled with the commercial finite element (FE) software COMSOL Multiphysics®, and the equation of motion is expressed as
K iC M q f , 2
(10)
where, K, C and M are the stiffness, damping and mass matrices, respectivly; q is nodal degree of T
p freedoms (DOFs); f is nodal forces. In this study, the DOFs are p in porous lining domain and T
p a in air domain, with the superscript T denoting matrix transpose. The porous material is modelled as a common fluid with complex density and complex elastic modulus, and the effect of perforated panel is represented by its impedance. 8
The short segment is meshed with structured quadratic lagrangian elements, where one element is used along the axial direction. Introducing the dynamic stiffness matrix D K iC 2M , the equation of motion can be expressed as Dq f . As shown in Fig. 1(a), the nodes in the short segment can be divided into left cross-sectional (L), internal (I) and right cross-sectional (R) nodes. Dividing the nodal DOFs and corresponding nodal forces into L, I and R parts, the equation of motion is partitioned as, DII DLI DRI
DIL DLL DRL
D IR q I 0 D LR q L f L . D RR q R f R
(11)
Eliminating qI in Eq.(11), the equation of motion for the left and right cross-section DOFs are obtained as,
DLL D RL
DLR q L f L , DRR q R f R
(12)
with the expression of matrix elements DLL, DLR, DRL and DRR provided in Appendix B. Re-ordering Eq. (12), the tranfer matrix formulation from the left cross-section to the right cross-section is obtained
q q T L R , fL fR
(13)
where the elements in matrix T are displayed in Appendix B. Combining Eq. (13) and periodic boundary condition
qR qL , fR fL
(14)
q q T L L . fL fL
(15)
the standard eigenvalue problem is formed as
The eigenvalue corresponds to the axial wavenumber kz of the eigenmode of the porous lined duct, with the relation
eik . z
(16)
The imaginary part of the wavenumber represents the attenuation of corresponding eigenmode, while real part implies the phase angle change. The first half of the eigenvector relates to the mode shape of the eigenmode. The analysis here concerns an infinite lined duct, so the axial wavenumbers are related to waves in a duct of infinite length. Since all the wavenumbers are complex numbers due to damping effect of porous materials, there is no rigorous criterion to determine the propagating, attenuating or evanescent modes. In this case, all the modes are simply divided into right-travelling and left-travelling waves with negative and positive imaginary parts of wavenumbers, respectively. In other words, the propagation 9
direction of the modes is determined by the requirement that all the modes are damped along the propagating path. When the Mach number is positive (flow direction is rightward), right-travelling modes propagate in the same direction as the uniform mean flow, while left-travelling modes propagate in the opposite direction. 3
Convergence analysis of the numerical scheme In this section, the calculated eigenvalues by WFE method are validated and the convergence rate of
eigenvalues is investigated. To compare with analytical results, a relatively simple geometry of a cylindrical duct without porous lining and perforated panel is considered. The radius of the circular crosssection is set as 0.1 m. For a circular duct with uniform flow, the axial wavenumbers of the cross-sectional eigenmodes can be analytically expressed as [35]
k
z ,m,n
Mk0 k02 1 M 2 kr2,m,n 1 M 2
, m, n 0,1, 2
,
(17)
where the superscripts + and − mean right-travelling and left-travelling modes, respectively; the subscripts r and z mean the axial and radial direction, respectively; the subscripts m and n are the mode index for circumferential and radial directions, respectively. kr,m,n is the n-th solution for the following eigen equation
dJ m kr r dr
0 at r r1 ,
(18)
where Jm is the m-th order Bessel function. The corresponding eigenmode Φm,n with the axial wavenumber kz,m,n can be expressed as
m,n r , J m kr ,m,n r eim .
(19)
The eigenmodes are referred to as (m,n) mode, with m and n being integers, where the first integer m represents the circumferential order and the second integer n represents the radial order. The relative error is defined as
kˆ k z
z
k z to show the convergence of the proposed method,
where kz is the value calculated by (17), and kˆz is the result calculated by the WFE method. The duct radius r1 is chosen as 0.1 m, and the calculated frequency in this section is chosen as 4000 Hz, if not specified. In our research, M < 1 is considered, so the calculated Mach number range is set as [0, 0.8].
10
3.1
Chosen of segment length The chosen of segment length Δ can affect the accuracy of the WFE method, so its effect is
investigated for different Mach numbers. In this section, the maximum element size is chosen as 1 mm, which will be shown to be good enough for convergence. The reletive error for right-travelling modes from analytical solution with M = 0, 0.5 and 0.8 are displayed in Fig. 2(a)-(c), respectively. It is found that for a fixed Mach number, the reletive error generally decreases with the decrease of Δ and then converges for both lower order modes (e.g. (1,1) mode) and higher order modes (e.g. (5,5) mode). However, compared with lower order modes, a smaller Δ is needed for the convergence of higher order modes. In the calculated example, Δ = 10-4 m can ensure the convergence of both lower order and higher order modes. In addition, the convergence patterns with Mach numbers from M = 0 to M = 0.8 are nearly the same. Therefore, in the following calculation, Δ = 10-4 m is chosen.
10
Relative error
10 10 10 10 10
0
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
-10
(a) M = 0
10-1
10-2
10-3 10-4 Segment length [m]
11
10-5
10
Relative error
10 10 10 10 10
0
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
-10
(b) M = 0.5
10-1
10
Relative error
10 10 10 10 10
10-2
10-3 10-4 Segment length [m]
10-5
0
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
-10
(c) M = 0.8
10-1
10-2
10-3 10-4 Segment length [m]
10-5
Fig. 2. Effect of segment length Δ on wavenumbers (relative error compared with analytical value) for right-travelling (1,1), (3,3) and (5,5) modes; (a) M = 0; (b) M = 0.5; (c) M = 0.8. 3.2
Convergence with mesh size The convergence of calculated wavenumbers with mesh size is investigated. The mesh size Se is
defined as the maximum element size of the mesh. When the mode expansion methods are used, convergence rate can be defined as the convergence with number of modes used [36, 37]. In FE analysis, the convergence with number of total nodes used is often adopted. However, the number of total nodes needed is dependent on the geometry of the calculation domain. Using the mesh size Se defined here, the convergence rate is independent on the geometric parameters, which shows more intrinsic convergence properties of the WFE method. 12
The relative error of calculated axial wavenumbers with the increase of element number are displayed in Fig. 3(a)-(c), respectively for M = 0, 0.5 and 0.8. In addition, a curve for trend of Se4 (relative value with reference value adjusted to show the trend in the figure) is also plotted for reference.
Relative error
10
10
10
10
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
Se4 10
Relative error
10
10
10
10
-10
(a) M = 0 8
4 2 Mesh size Se [mm]
1
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
Se4 10
-10
(b) M = 0.5 8
4 2 Mesh size Se [mm]
13
1
Relative error
10
10
10
10
(1,1) Right (3,3) Right (5,5) Right
-2
-4
-6
-8
Se4 10
-10
(c) M = 0.8 8
4 2 Mesh size Se [mm]
1
Fig. 3. Convergence of wavenumbers (relative error compared with analytical value) with mesh size Se for right-travelling (1,1), (3,3) and (5,5) modes, together with the trend of Se4 provided for reference; (a) M = 0; (b) M = 0.5; (c) M = 0.8. It is found that the relative error of calculated axial wavenumbers decreases with the decrease of mesh size Se, whose trend is parallel to the trend of Se4. Therefore, a convergence rate of Se4 for both lower order and higher order modes with Mach number ranging from 0 to 0.8 is realized with the WFE method. However, smaller mesh size is needed for higher order modes to achieve the same accuracy as lower order modes. The convergence rate at different frequencies is also investigated. The calculation example for (3,3) right-travelling mode at three different frequencies is shown in Fig. 4. It is found that the convergence rate of Se4 is kept up to 8000 Hz.
14
Fig. 4. Convergence of wavenumber with mesh size Se at three different frequencies, shown as relative error compared to analytical value, for the (3,3) right-travelling mode, together with the trend of Se4 provided for reference. 4
Effects of mean flow on the eigenmode characteristic of the lined ducts The eigenmodes of the porous lined duct with mean flow can be obtained with the WFE method, as
described Section 2. In this section, the cross-sectional geometry of the lined duct is set as circular for easier analysis, as depicted in Fig. 5(b).
Γinlet A U0
B
I
pi
Γ1 z 0 C r
L0
Perforated panel
Γ2 Air
r2
II
F
E L (a) Axial-sectional view
Porous material r1
Air
r2 (b) Cross-sectional view 15
III L0
Porous material D
r1
Γoutlet H G
Fig. 5. Geometry of the porous lined duct with circular cross-section; (a) axial sectional view; (b) crosssectional view. 4.1
Eigenvalue and eigenmode In this section, the eigenmodes and corresponding wavenumbers are calculated for the lined duct,
with the geometry and material parameters described in Table 2. The calculated axial wavenumbers for the eigenmodes of the porous lined duct are shown in Fig. 6. The Mach numbers is set as M = 0.2, the circumferential order is set as m = 3, and the frequency is chosen as 4000 Hz. It is found that the wavenumbers are not symmetric about the origin because of the uniform mean flow. 400 Right Left
Imag(k) [1/m]
200
lining-borne mode
air-borne mode
0
-200 air-borne mode
-400 -100
-50
lining-borne mode
0 Real(k) [1/m]
50
100
Fig. 6. Axial wavenumbers at 4000 Hz with M = 0.2 and m = 3; right- and left-travelling modes are marked with blue asterisk and red cross marks respectively; air-borne and lining-borne modes are marked respectively.
16
(a) air-borne modes
(b) Lining-borne modes
17
Fig. 7. Mode shapes of first 3 eigenmodes at 4000 Hz with M = 0.2 and m = 3; (a) right- and lefttravelling air-borne (3,0)a, (3,1)a and (3,2)a modes; (b) right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes. In addition, in both right-travelling modes and left-travelling modes, the eigenvalues can be divided into air-borne modes and lining-borne modes. The mode shapes for pressure field of first 3 air-borne modes are shown in Fig. 7(a). It can be seen that the mode shapes of air-borne modes for both righttravelling and left-travelling waves are nearly the same, which mainly propagate in the air domain, while the pressure in the porous material domain is relatively small. In addition, with the increase of radial order, the number of cross-zero points in both air-domain and porous material domain increases. The mode shapes for pressure field of first 3 lining-borne modes are shown in Fig. 7(b). In contrast to the air-borne modes, the mode shapes of lining-borne modes are different for right-travelling and lefttravelling waves, when there is uniform mean flow. For the right-travelling modes, in other words, the direction of the wave propagation is the same as flow direction, the wave mainly propagates in lining domain for lining-borne modes. For the left-travelling modes, which travels in negative flow direction, some of the lining-borne modes propagate only in lining domain (e.g. (3,0)l mode), while some in both lining domain and air domain (e.g. (3,1)l mode). In other words, the main characteristic for the liningborne modes is that they can all propagate in porous lining domain, while some of them also propagate in the air domain but some do not. Therefore, the modes for left-travelling wave are more influenced by the uniform mean flow, which will be further described in the next section. As the mode properties may change along with frequency, the mode classification is conducted according to the calculation result at 4000 Hz with Mach number M = 0.2. The influence of frequency on the eigenmodes is investigated by the kinetic energy ratio RKE in two domains, which is inspired by Ref. [38], and defined as RKE
Ekl , Eka
(20)
where Ekl and Eka are averaged kinetic energy in lining domain and air domain, respectively. For the airborne mode, RKE is generally much larger than 1, while for lining-borne mode, RKE is around 1 or much smaller than 1. In the lining domain, there is no mean flow, so the kinetic energy can be simply calculated by the kinetic energy with acoustic particle velocity, as
Ekl
1 2 Re c v dS 4 lining domian
Sl
18
,
(21)
where v is the acoustic particle velocity and Sl is the total area of lining domain. In the air domain, uniform mean flow exists, so there is no strict criterion to separate kinetic energy from total acoustic energy. Moreover, it is still a subject of discussion that which part of the field is to be called acoustic energy with the existence of flow field [39]. Therefore, the analog of definition in Eq. (21) is used to represent the kinetic energy in air domain Eka , which is defined as 1 2 0 v dS 4 Ekl air domian , Sa
(22)
where v is the acoustic particle velocity and Sa is the total area of air domain. The evolution of kinetic energy ratio with frequency for different modes is displayed in Fig. 8(a)-(d). It is found that with the change of frequency, higher order right- and left-travelling air-borne modes, e.g. (3,1)a mode and (3,2)a mode, generally keep their properties as air-borne modes. For both right- and lefttravelling air-borne (3,0)a mode, it performs like a lining mode in the low-frequency range (with RKE around 1), and gradually evolves to the air mode in the high-frequency range. The evolution of liningborne modes is different from air-borne modes. For both right- and left-travelling lining-borne (3,0)l mode, it keeps as lining-borne mode with the change of frequency. On the contrary, for higher order right- and left-travelling lining-borne modes, e.g. (3,1)l mode and (3,2)l mode, they show the properties of air-borne modes in the low-frequency range, and evolve to lining-borne modes with the increase of frequency.
19
20
Fig. 8. Evolution of kinetic energy ratio RKE with frequency for different modes; (a) right-travelling airborne (3,0)a, (3,1)a and (3,2)a modes; (b) left-travelling air-borne mode (3,0)a, (3,1)a and (3,2)a modes; (c) right-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes; (d) left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes. 4.2
Effect of Mach number on eigenmodes The effect of Mach number on the axial wavenumbers and mode shapes are described in this section.
The range of Mach number is set as [0, 0.5], and frequency is chosen as 4000Hz. The wavenumber evolution with the increase of Mach number is demonstrated in Fig. 9. It is found that, the effect of Mach number on air-borne modes and lining-borne modes are different. The change of Mach number will make greater changes of wavenumbers for air-born modes, compared with its effect on lining-borne modes. For the left-travelling air-borne modes, the increase of Mach number will mainly increase the real part of the wavenumber. It can be easily understood that increase of Mach number will effectively decrease the speed of sound for the waves travelling in negative flow direction, which in turn increases the real part of the wavenumber. Compared with real part, the effect of Mach number on imaginary part is small. For the right-travelling air-borne modes, the real part of some lower order modes ((3,0)a mode in this calculation example) will decrease with the increase of Mach number, which can be explained by the effectively increased speed of sound; while the real part of higher order modes (other modes in this calculation example) will increase with the increase of Mach number, which is due to the effect of flow convection. Generally speaking, the increase of Mach number will shift the wavenumbers of air-borne modes to the left on the complex plane of wavenumbers. 21
Fig. 9. Effect of Mach number on axial wavenumbers for right-travelling (solid line) and left-travelling (dash line) modes with M = 0~0.5; circular markers for M = 0 and triangular markers for M = 0.5. For right-travelling lining-borne modes, the change of Mach number has little effect on the wavenumber. For some particular left-travelling lining-borne mode (e.g. (3,2)l in this calculation example), the behaviour of the wavenumber with the change of Mach number is more like an air-borne mode. It is further supported by the its mode shape, which will be described in the following. The aforementioned result is summarized in Table 1 for reference. Table 1 Axial wavenumbers of the calculation example at 4000Hz. M=0, Right
M=0, Left
M=0.5, Right
M=0.5, Left
(3,0)a
54.83-2.25i
-54.83+2.25i
34.09-3.74i
-120.74+3.70i
(3,0)l
78.54-10.10i
-78.54+10.10i
78.45-10.26i
-78.74+9.84i
(3,1)a
2.18-38.95i
-2.18+38.95i
-19.95-15.41i
-46.67+29.97i
(3,1)l
69.31-10.75i
-69.31+10.75i
68.34-12.70i
-72.93+9.16i
(3,2)a
0.73-89.20i
-0.73+89.20i
-40.44-85.32i
-45.63+96.94i
(3,2)l
42.59-16.29i
-42.59+16.29i
41.98-20.83i
-57.93+15.93i
(3,3)a
0.42-127.91i
-0.42+127.91i
-43.30-133.00i
-44.30+144.35i
(3,3)l
11.80-60.08i
-11.80+60.08i
15.92-63.69i
-17.66+59.81i
22
(a) M = 0
(b) M = 0.5 Fig. 10. Effect of Mach number on mode shapes for right- and left-travelling air-borne (3,0)a, (3,1)a and (3,2)a modes at 4000 Hz; (a) M = 0; (b) M = 0.5. 23
The mode shapes of air-borne modes are shown in Fig. 10(a) and (b), respectively for M = 0 and M = 0.5. It is found that, for M = 0, in other words, there is no mean flow in the duct, the mode shapes for right-travelling and left-travelling waves are exactly the same. When there is uniform mean flow with M = 0.5, the mode shapes are changed a little bit from the case with M = 0. However, for both left-travelling and right-travelling waves, the effect of Mach number on the mode shapes of air-borne modes are small. The mode shapes of lining-borne modes are shown in Fig. 11(a) and (b), respectively for M = 0 and M = 0.5. It is found that for right-travelling modes, the increased Mach number slightly changes the mode shape. For left-travelling modes, the mode shapes are strongly affected by the change of Mach number. In the calculation example, the (3,0)l mode mainly propagates in porous lining domain with M = 0, while it can propagate in both porous lining domain and air domain when the Mach number increases to 0.5. For the (3,2)l mode, it mainly propagates in porous lining domain with M = 0, while it changes to performs like an air-borne mode M = 0.5, which is in accordance with the trend of corresponding wavenumber.
(a) M = 0
24
(b) M = 0.5 Fig. 11. Effect of Mach number on mode shapes for right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes at 4000 Hz; (a) M = 0; (b) M = 0.5. It is summarized that, for air-borne modes, Mach number will strongly affect their wavenumbers, but its effect on the mode shapes is small. On the contrary, for lining-borne modes, Mach number has small effect on wavenumbers, but its effect on mode shapes is strong, even the mode type can be changed from a lining-borne mode to an air-borne mode for some particular left-travelling mode. 4.3
Effect of Mach number on transmission loss The transmission loss (TL) of a lined duct with mean flow is calculated with the mode matching
method [1], which is described in detail in Appendix A, to further validate the calculated eigenmodes and corresponding axial wavenumbers. In addition, the effect of Mach number on the TL is also analysed. In this calculation example, the plane wave incidence condition is applied. The procedure of the mode matching scheme is briefly described here for completeness. The axialsectional view of calculation geometry is shown in Fig. 5 (a). Regions I and III are air ducts with radius r1 and sound wave comes from the left. Region II is porous lined duct with radius r2, while the air domain has the same radius as Region I and III. The length of the lined part is L, and the length of the air duct is L0, which is chosen to be the same as L in this calculation example. The waves inside each region (I, II and III) are expressed as the superposition of eigenmodes of the right-travelling and left-travelling waves with the unknown mode amplitudes to be solved. Applying the boundary condition on the two interfaces 25
Γ1 and Γ2 (as shown in Fig. 5 (a)), the equation for the unknow mode amplitudes can be obtained, and the TL can be then obtained through calculated mode amplitudes. For Region II, the eigenmodes and wavenumbers are obtained with the WFE method. For Regions I and III, analytical solutions for the eigenmode and axial wavenumber exist, described as Eqs. (18)-(19). The axial wavenumbers and corresponding eigenmodes are sorted out by the absolute value of the imaginary part of wavenumbers in ascending order, ensuring that the least attenuated modes are included in the mode matching scheme. The parameters of the lined duct geometry and lining porous materials as well as perforated panel are listed in Table 2. Table 2 Geometry and material parameters. Lined duct geometry
Perforated panel
Porous material
Length L = L0 [mm]
300
Thickness th [mm]
1
Radius of air duct r1 [mm]
100
Hole diameter d [mm]
1
Radius of lined duct r2 [mm]
100
Porosity σ
5%
Flow resistivity Rf [Pa·s/m2]
5000
The TL of the lined duct with M = 0.2 is shown in Fig. 12 (solid line), together with the result from full FE simulation (circular points). It is found that, the TL calculated with eigenmodes matches well with that from full FE simulation. In this case, the eigenmodes and eigenvalues of the porous lined duct with uniform mean flow are validated.
12
TL [dB]
10 8 6 Mode matching Full FE
4 2
0
500
1000 Frequency [Hz]
1500
2000
Fig. 12. Validation of transmission loss (TL) calculated by mode matching with full FE simulation for M = 0.2.
26
Fig. 13. Effect of Mach number on transmission loss (TL) of lined duct with three Mach numbers: M = 0, M = -0.2 and M = 0.2. The effect of Mach numbers on TL is investigated and shown in Fig. 13. It is found that, when the Mach number is negative, the TL is increased for both low-frequency range and high-frequency range. However, the peak value for TL is decreased with negative Mach number. In addition, the peak frequency is nearly not shifted. When the Mach number is positive, TL is generally smaller than the case of M = 0. In other words, TL will generally increase with up-flow (M = -0.2), except for frequencies around the peak; while it will generally decrease with down-flow (M = 0.2). In both cases, the peak frequency does not show significant change. The effect of flow resistivity Rf on the TL is investigated, and the result is shown in Fig. 14 for the calculated example with M = 0.2. It is found that with the increase of Rf, the peak value of TL decreases and the high frequency TL increases. In addition, the peak frequency shifts to higher frequency range with the increase of Rf. These characteristics are in accordance with the case of no mean flow [40].
27
Fig. 14. Effect of flow resistivity Rf (5000, 10000 and 20000 Pa·s/m2) on transmission loss (TL) of lined duct with M=0.2. 5
Conclusions In this paper, the effect of uniform mean flow on the eigenmodes of lined ducts is analyzed with a
combined wave and finite element (WFE) method. The duct is lined with bulk-reactive porous materials, which is separated from the flow channel by a perforated panel. The proposed method is validated by the analytical solution and the convergence rate of the method is discussed. It is found that the method is applicable for Mach number M at least up to 0.8. With properly chosen of segment length , the WFE method can reach the convergence rate of Se4 for eigenvalues with M ≤ 0.8, where Se is the mesh size. Nevertheless, smaller mesh size is needed for higher order modes to achieve the same accuracy as lower order modes. With porous lining and perforated panel, the eigenmodes can be divided into air-borne modes and lining-borne modes for both right-travelling and left-travelling waves. The air-borne modes propagate mostly in the air domain, while the lining-borne modes can propagate in both air domain and porous domain. The effect of Mach number on the axial wavenumbers and mode shapes are analysed. It is found that, Mach number will strongly affect the wavenumbers of air-borne modes. The increase of Mach number will shift the wavenumbers of air-borne mode to the left on the complex plane of wavenumbers. However, the effect of Mach number on mode shapes of air-borne modes is small. On the contrary, for the liningborne modes, Mach number has small effect on wavenumbers, but its effect on mode shapes is strong,
28
even the mode type can be changed from a lining-borne mode to an air-borne mode for some lefttravelling modes. The transmission loss (TL) of a typical porous lined duct with three typical Mach numbers M = 0, 0.2 and -0.2 is calculated with the eigenmodes by the mode matching method. The calculated TL matches well with the result from full finite element simulation, which further validates the calculated eigenmodes. It is found that up-flow (M = -0.2) will generally increase TL of lined duct, compared with the case of no flow, except for frequencies around the peak; while down-flow (M = 0.2) will generally decrease TL. In both cases of up-flow and down-flow, the peak frequency does not show significant change. Acknowledgment Part of the work was accomplished when the first author was the Post-Doctoral Fellow in The University of Hong Kong. This project is supported by National Natural Science Foundation of China (Grant No. 51775467).
AUTHOR DECLARATION We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
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Structural Dynamics (RASD), Journal of Physics: Conference Series, 15–17 April 2019, Valpre, Lyon, France, 2019, pp. 012056. [39] S.W. Rienstra, A. Hirschberg, An introduction to acoustics, Eindhoven University of Technology, 2018. [40] I.L. Vér, L.L. Beranek, Noise and vibration control engineering: principles and applications, 2 ed., Wiley Online Library, 2006.
32
Appendix A.
Mode matching sheme [1] The calculation geometry setting is shown in Fig. 5 (a). The waves inside each region (I, II and III)
are expressed as the superposition of eigenmodes of the right-travelling and left-travelling waves with the unknown mode amplitudes to be solved. For example, the air domain pressure in Region II is expressed as Nm
p a ,II r , z AnII, pII,a ,n r e n 1
izk zII,n,
AnII, pII,a ,n r e
izk zII,n,
,0 r r , 1
(A-1)
where, the superscript a denotes the air part; the superscript II denotes region II; symbols + and – denote the right-travelling and left-travelling waves, respectively, pII,a , n and pII,a , n are air domain pressure parts of the corresponding n-th eigenmodes ; k zII,,n and k zII,,n are the axial wavenumbers; AnII, and AnII, are the unknown mode amplitudes , Nm is the number of eigenmodes used. Changing the above equation into the matrix form gives
p a ,II r , z pII,a r EII, z AII, pII,a r EII, z AII, ,0 r r1 ,
(A-2)
II,a / r II,a / r , II,a / r , , II,a / r p ,2 p , Nm p ,1 p II , / II, / II , / II , / izk z diag eizk1 ,eizk2 , ,e Nm . E A II, / AII, / , AII, / , , AII, / T 2 Nm 1
(A-3)
where,
Applying the continuity of pressure and normal component of particle displacement on air part of the two interfaces Γ1 and Γ2 (0
(A-4)
A III, A III, X2 E X 2 II, Y2 EY 2 II, , A A
(A-5)
with
PaI, X1 U I,a 0
PaI, PaII, I, U II, a , Y1 U a 0 WpII,
PaIII, PaII, U aII, , X2 U III, a 0 WpII, 33
PaIII, PaII, III, U II, a , Y2 U a 0 WpII,
PaII, U aII, , WpII,
(A-6)
EIII, EX 2 0
EIII, 0 , E z Y2 EII, 0
0 . EII,
(A-7)
Al,s (l = I, II, III and s = +,–) is a column vector with the length of Nm, containing the right-travelling or left-travelling mode amplitudes. El,t (l = I, II, III and s = +,–) is a diagonal matrix, which is defined as
El , s diag e
izk zl ,,1s
,e
izk zl ,,2s
,e
izk zl ,,3s
, ,e
izk zl ,,sN
m
. The components in matrix X , Y and X , Y are scattering 1
1
2
2
matrix calculated by boundary integration on the interface, which is defined as follows. On the left boundary Γ1, † r1 PaI, / ψ pa pI,a / rdr 0 † r II, / 1 ψ p a pII,a / rdr Pa 0 † r1 I, / I, / U a ψ uza uza rdr , 0 † r1 II, / U II, / ψ 0 uza uza rdr a † II , / r2 ψ u p uIIp, / rdr Wp z z r1
(A-8)
with ψ pa pI,a , ψua uII,a , ψu p uII,p , and the superscript † referring to take transpose conjugate of a z
z
z
z
vector. On the right boundary Γ2, † r1 III, / Pa ψ pa pIII,a / rdr 0 † r1 II, / ψ p a pII,a / rdr Pa 0 † r III, / 1 / ψ u a uIII, rdr , a U a z z 0 † r1 II, / U II, / ψ 0 uza uza rdr a † II , / r2 ψ u p uIIp, / rdr Wp z z r1
(A-9)
with ψ pa pIII,a , ψua uII,a , ψu p uII,p . z
z
z
z
In the present calculation example, a plane wave incidence with unit amplitude is assumed, so that AI,+ = [1,0,0…,0]T. On the outlet interface, a non-reflecting boundary condition is applied with AIII,– = 34
[0,0,0…,0]T. The unknown mode amplitudes are obtained by solving Eqs. (A-4) and (A-5). The TL of the lined duct is then found as
TL 20log10 A1III, . B.
(A-10)
Matrix in the eigenvalue equation In Eq. (12), the new partitioned dynamic stiffness matrixs DLL, DLR, DRL and DRR are expressed as D LL D LL D LI DII1D IL 1 D LR D LR D LI D II D IR . 1 D D D D D RL RL RI II IL 1 D RR D RR D RI D II D IR
(B-1)
In Eq. (13), the elements in the transfer matrix T are expressed as
DLR1 DLL T 1 DRL DRR DLR DLL
35
DLR1 . DRR DLR1
(B-2)
Figure captions Fig. 1.
Geometry of the porous lined duct with arbitrary cross-section containing a uniform mean flow U0; (a) axial-sectional view with axial direction set as z axis; (b) cross-sectional view with 2D coordinate system; (c) 3D view for a short segment with length .
Fig. 2.
Effect of segment length Δ on wavenumbers (relative error compared with analytical value) for right-travelling (1,1), (3,3) and (5,5) modes; (a) M = 0; (b) M = 0.5; (c) M = 0.8.
Fig. 3.
Convergence of wavenumbers (relative error compared with analytical value) with mesh size Se for right-travelling (1,1), (3,3) and (5,5) modes, together with the trend of Se4 provided for reference; (a) M = 0; (b) M = 0.5; (c) M = 0.8.
Fig. 4.
Convergence of wavenumber with mesh size Se at three different frequencies, shown as relative error compared to analytical value, for the (3,3) right-travelling mode, together with the trend of Se4 provided for reference
Fig. 5.
Geometry of the porous lined duct with circular cross-section; (a) axial sectional view; (b) cross-sectional view.
Fig. 6.
Axial wavenumbers at 4000 Hz with M = 0.2 and m = 3; right- and left-travelling modes are marked with blue asterisk and red cross marks respectively; air-borne and lining-borne modes are marked respectively.
Fig. 7.
Mode shapes of first 3 eigenmodes at 4000 Hz with M = 0.2 and m = 3; (a) right- and lefttravelling air-borne (3,0)a, (3,1)a and (3,2)a modes; (b) right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes.
Fig. 8.
Evolution of kinetic energy ratio RKE with frequency for different modes; (a) right-travelling air-borne (3,0)a, (3,1)a and (3,2)a modes; (b) left-travelling air-borne mode (3,0)a, (3,1)a and (3,2)a modes; (c) right-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes; (d) left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes.
Fig. 9.
Effect of Mach number on axial wavenumbers for right-travelling (solid line) and left-travelling (dash line) modes with M = 0~0.5; circular markers for M = 0 and triangular markers for M = 0.5.
Fig. 10.
Effect of Mach number on mode shapes for right- and left-travelling air-borne (3,0)a, (3,1)a and (3,2)a modes at 4000 Hz; (a) M = 0; (b) M = 0.5.
Fig. 11.
Effect of Mach number on mode shapes for right- and left-travelling lining-borne (3,0)l, (3,1)l and (3,2)l modes at 4000 Hz; (a) M = 0; (b) M = 0.5.
36
Fig. 12.
Validation of transmission loss (TL) calculated by mode matching with full FE simulation for M = 0.2.
Fig. 13.
Effect of Mach number on transmission loss (TL) of lined duct with three Mach numbers: M = 0, M = -0.2 and M = 0.2.
Fig. 14.
Effect of flow resistivity Rf (5000, 10000 and 20000 Pa·s/m2) on transmission loss (TL) of lined duct with M = 0.2.
Table captions Table 1
Axial wavenumbers of the calculation example at 4000Hz.
Table 2
Geometry and material parameters. Table 1 Axial wavenumbers of the calculation example at 4000Hz. M=0, Right
M=0, Left
M=0.5, Right
M=0.5, Left
(3,0)a
54.83-2.25i
-54.83+2.25i
34.09-3.74i
-120.74+3.70i
(3,0)l
78.54-10.10i
-78.54+10.10i
78.45-10.26i
-78.74+9.84i
(3,1)a
2.18-38.95i
-2.18+38.95i
-19.95-15.41i
-46.67+29.97i
(3,1)l
69.31-10.75i
-69.31+10.75i
68.34-12.70i
-72.93+9.16i
(3,2)a
0.73-89.20i
-0.73+89.20i
-40.44-85.32i
-45.63+96.94i
(3,2)l
42.59-16.29i
-42.59+16.29i
41.98-20.83i
-57.93+15.93i
(3,3)a
0.42-127.91i
-0.42+127.91i
-43.30-133.00i
-44.30+144.35i
(3,3)l
11.80-60.08i
-11.80+60.08i
15.92-63.69i
-17.66+59.81i
Table 2 Geometry and material parameters. Lined duct geometry
Perforated panel
Porous material
Length L = L0 [mm]
300
Thickness th [mm]
1
Radius of air duct r1 [mm]
100
Hole diameter d [mm]
1
Radius of lined duct r2 [mm]
100
Porosity σ
5%
37
Flow resistivity Rf [Pa·s/m2]
5000
Graphical_abstract
38