Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 2, pp. 313-325, Warsaw 2013 NUMERICAL CHARACTERIZATION OF AN AXISYMMETRIC LINED DUCT WITH FLOW USING THE MULTIMODAL SCATTERING MATRIX Mohamed Taktak, Mohamed Ali Majdoub, Mohamed Haddar University of Sfax, National School of Engineers of Sfax, Unit of Mechanics, Modelling and Production, Sfax, Tunisia e-mail: mohamed.taktak@fss.rnu.tn Mabrouk Bentahar University of Technology of Compiègne, Roberval Laboratory UMR UTC-CNRS no. 6253, Compiègne Cedex, France In this paper, the development of a numericalmethod to compute themultimodal scattering matrix of a lined duct in the presence of flow is presented. This method is based on the use of the convectedHelmholtz equation and the addition ofmodal pressures at duct boundaries as additional degrees of freedom of the system. The boundary effects at the inlet and outlet of the finite waveguide are neglected.The choice of thismatrix is justified by the fact that it representsan intrinsic characterizationofaduct system.Thevalidationof theproposedfinite element is done by a comparison with the analytical formulation for simple cases of ducts. Then, the numerical coefficients of the scattering matrix of a lined duct and its acoustic power attenuation are computed for several flow velocities to evaluate the flow effect. Key words: lined duct, scattering matrix, mean flow 1. Introduction The characterization of the acoustic behavior of aircraft engines is an important tool used by engine designers to reduce the noise inside such systems and radiated from them. These engines are generally presentedas awave guide composed of anacoustic source anda series of a rigidwall and linedducts.Tocharacterize thesewave guide systems, somespecificmatrices areused suchas themobilitymatrix as used byPierce (1981), transfermatrix, see To andDoige (1979a,b), Lung and Doige (1983), Munjal (1987), Peat (1988) and Craggs (1989), reflection matrix presented in Akoum andVille (1998) and Sitel et al. (2003), transmissionmatrix see Sitel et al. (2003) or scattering matrix see Abom (1991), Leroux et al. (2003), Bi et al. (2006) and Sitel et al. (2006) matrices. In a previous work, Taktak et al. (2010) developed the multimodal scattering matrix of a lined duct to characterize an axisymmetric rigid wall – lined – rigid wall duct simulating an aircraft engine without flow. In fact, this matrix represents an intrinsic characterization of the duct element independently of the upstream and downstream conditions: it depends only on acoustics and geometrical duct features and provides a complete description of the modal reflection, transmission and conversion of the duct element. Thismatrix is also used to evaluate the efficiency of the duct by computing its acoustic power attenuation as presented byAurégan and Starobinski (1998) and Taktak et al. (2010). In that latter work, the scattering matrix was used to evaluate the efficiency of a lined duct and to characterize duct edges by calculation of its acoustic impedance without flow. But in a real engine the flow is present and has an important effect on the acoustic behavior of liners. For this rreason, a method based on the finite element method to compute themultimodal scatteringmatrix of a linedduct in thepresence of a uniform flowwith aMach number smaller than unity is presented in this paper. Thismatrix is then used to characterize the acoustic performance of the studied duct by computing its acoustic power attenuation and to evaluate the flow effects on these parameters (scattering coefficients and acoustic attenuation). 314 M. Taktak et al. In this paper, the studied problem is presented in Section 2. Then, the finite elementmethod to compute the numerical multimodal scattering matrix with flow of a lined duct is presented in Section 3. Section 4 presents the computation of the acoustic power attenuation from the scattering matrix. Results of the proposed numerical method are presented and discussed in Section 5 to evaluate the flow effect. 2. Description of the physical problem The studied duct is cylindrical. Figure 1 presents its symmetric part. It does not present a sudden section change but an impedance discontinuity caused by the liner which is supposed to be locally reacting ismodeled by its acoustic impedance Z. Ω is the acoustic domain inside the duct. The edge of the studied duct is composed of four parts: the rigid wall duct part ΓWD, the lined duct part ΓLD, the left transversal boundary ΓL and the right transversal boundary ΓR. ΓWD, ΓLR, ΓL and ΓR are characterized respectively by their normal vectors nWD, nLD, nL and nR. A uniformflowwith aMach number smaller than unity is present in this ductmodeled by the vector M0 defined as M0 = (U0 c ) = (U0z c ) =M0z (2.1) where M0 is theMach number, U0 is the flow velocity, c is the sound velocity and z is the duct axis. The objective of this work is the characterization of an industrial duct composed of a rigid wall and lined parts and the evaluation of its efficiency as well as the flow effect on the acoustic behavior of this duct. This is obtained by using the multimodal scattering matrix, from which the acoustic power attenuation is deduced. The methodology of numerical computation of this matrix as well as of the acoustic attenuation is presented in the next sections. 3. Computation of the multimodal scattering matrix 3.1. Definition of the scattering matrix The scattering matrix SN×N of the duct element relates the outcoming pressure wa- ves array Pout2N = [P I− 00 , . . . ,P I− PQ,P II+ 00 , . . . ,P II+ PQ ] T N to the incoming pressure waves array Pin2N = [P I+ 00 , . . . ,P I+ PQ,P II− 00 , . . . ,P II− PQ ] T N (Fig. 1) as follows, see Taktak et al. (2010) P out 2N =S2N×2NP in 2N = [ R + N×N T + N×N T − N×N R − N×N ] 2N×2N P in 2N (3.1) where PI+mn and P I− mn are themodal pressure coefficients associated to the (m,n)mode traveling, respectively, in the positive and the negative direction in region I, PII+mn and P II− mn are respecti- vely themodal pressure coefficients associated to the (m,n)mode traveling, respectively, in the positive and the negative direction in region II (Fig. 1). m and n are, respectively, the azimuthal and the radial mode numbers. N is the number of modes in both cross sections, P and Q are, respectively, the angular and radial wave numbers associated to the N-th propagating mode (m¬P and n¬Q). 3.2. Governing equations The studied duct is axisymmetric. The boundary effects at the inlet and outlet of the duct are neglected. The acoustic pressure p in the duct is the solution of the system containing Numerical characterization of an axisymmetric lined duct... 315 Fig. 1. Schematic of the theoretical model for the computation of the multimodal scatteringmatrix the convected Helmholtz equation with boundaries conditions at ΓWD (rigid wall duct part) and ΓLD (lined duct part) ∆p+k2p+ 2iω c (M0 ·∇p)−M0 ·∇(M0 ·∇p)= 0 (Ω) Z ∂p ∂nLD = ρ0 iω ( −iω+U0 ∂ ∂z )2 (p) (ΓLD) ∂p ∂nWD =0 (ΓWD) (3.2) where ∆ is the Laplacian operator; k is the total wave number, ρ0 is the density and ω is the pulsation. ∇= 〈∂/∂r, im/r,∂/∂z〉T is themodified gradient for axisymmetric problems. χmn is the n-th root satisfying the radial hard-boundary condition on the rigid wall of the main duct. The acoustic pressure fields at the left section ΓL and the right section ΓR (Fig. 1) are given as follows pL = Nr ∑ n ( PI+mne ik + mn(z−zL)+PI−mne ik − mn(z−zL) ) Jm (χmn a r ) pR = Nr ∑ n ( PII+mn e ik+mn(z−zR)+PII−mn e ik−mn(z−zR) ) Jm (χmn a r ) (3.3) with Nr being the number of radial modes. zL and zR are, respectively, the axial position of the left and right boundaries, Jm is the Bessel function of the first kind of the order m, a is the duct radius, r is the radial variable. k±mn are the axial wave numbers associated with the (m,n) mode and defined as k±mn = −M0k± √ k2− (1−M20)k 2 t 1−M20 (3.4) where kt is the transverse wave number. The sign “+” means that the axial wave number is calculated in the same direction of the flow, the sign “−” means that the axial wave number is calculated in the opposite direction of the flow. 3.3. Variational formulation To solve problem (3.2), the finite element method is used. Theweak variational formulation of this problem is written as follows Π = ∫ Ω −(∇q ·∇p)r dΩ+ 1 c2 ∫ Ω (iωq+U0 ·∇q)(−iωp+U0 ·∇p)r dΩ + ∫ ∪Γi [ q ∂p ∂ni − 1 c2 U0 ·niq ( −iω+U0 ∂ ∂ni ) (p)r dΓi =0 (3.5) 316 M. Taktak et al. where p and q are, respectively, the acoustic pressure in the duct and the test function. dΩ = drdz is the surface element. ∪Γi present the whole boundaries (i = LD – lined part, i = L – left, i = R – right). The third integral includes boundary conditions. This integral is composed of three parts: — Lined part ΓLD ∫ ΓLD [ q ∂p ∂nLD − 1 c2 U0 ·nLDq ( −iω+U0 ∂ ∂nLD ) (p) ] r dΓLD =−ρ0ω 2 ∫ ΓLD q p iωZ r dΓLD −2iωρ0U0 ∫ ΓLD q ∂ ∂z ( p iωZ ) r dΓLD −ρ0U 2 0 ∫ ΓLD ∂q ∂z ∂ ∂z ( p iωZ ) r dΓLD +ρ0U 2 0 [ rq ∂ ∂z ( p iωZ )]LLD (3.6) with LLD being the lined part length. — Left boundary ΓL ∫ ΓL [ q ∂p ∂nL − 1 c2 U0 ·nLq ( −iω+U0 ∂ ∂nL ) (p) ] r dΓL = Nr ∑ n=1 inL [ (1+M20)(k + mnP I+ mn+k − mnP I− mn)−kM0(P I+ mn+P I− mn) ] ∫ ΓL qJm (χmn a r ) r dΓL (3.7) —Right boundary ΓR ∫ ΓR [ q ∂p ∂nR − 1 c2 U0 ·nRq ( −iω+U0 ∂ ∂nR ) (p) ] r dΓR = Nr ∑ n=1 inR [ (1+M20)(k + mnP II+ mn +k − mnP II− mn )−kM0(P II+ mn +P II− mn ) ] ∫ ΓR qJm (χmn a r ) r dΓR (3.8) Theuse ofmodal decomposition at the boundaries ΓL and ΓR inEq. (3.3) introduces themodal pressures as additional degrees of freedom of the model. It is necessary to complete Eqs. (3.5), (3.6) and (3.7) with more equations to obtain a well posed problem. This is done by supposing that pressures at ΓL and ΓR can be obtained by the projection of the acoustic field over the eigenfunctions of the rigid wall duct ∫ ΓL pJm (χmn a r ) dΓL =(P I+ mn+P I− mn) ∫ ΓL Jm (χmn a r )2 r dΓL ∫ ΓR pJm (χmn a r ) dΓR =(P II+ mn +P II− mn ) ∫ ΓR Jm (χmn a r )2 r dΓR (3.9) 3.4. Finite element discritization To solve the proposed problem, the domain (Ω) is discretized with triangular finite elements while the edges are meshed by two node finite elements. The computation of integrals of Eq. (3.4) is made by the summation over the finite elements number of elementary integrals (Dhatt and Touzout, 1989) Numerical characterization of an axisymmetric lined duct... 317 Ie1 = ∫ Ωe −(∇q ·∇p)r dΩe+ 1 c2 ∫ Ωe (iωq+U0 ·∇q)(−iωp+U0 ·∇p)r dΩe Ie2 =−ρ0ω 2 ∫ Γe q p iωZ r dΓe−2iωρ0M0 ∫ Γe q ∂ ∂z ( p iωZ ) r dΓe−ρ0M 2 0 ∫ Γe ∂q ∂z ∂ ∂z ( p iωZ ) r dΓe Ie3 = ρ0M 2 0 [ rq ∂ ∂z ( p iωZ )]LLD Ie4 = Nr ∑ n=1 inL [ (1+M20)(k + mnP I+ mn+k − mnP I− mn)−kM0(P I+ mn+P I− mn) ] ∫ Γe qJm (χmn a r ) r dΓe Ie5 = Nr ∑ n=1 inR [ (1+M20)(k + mnP II+ mn +k − mnP II− mn )−kM0(P II+ mn +P II− mn ) ] ∫ Γe qJm (χmn a r ) r dΓe (3.10) The computation of integrals (3.9) is obtainedby the summation over the finite elements number of elementary integrals Ie6 = ∫ Γe pJm (χmn a r ) r dΓe− (P I+ mn+P I− mn) ∫ Γe Jm (χmn a r )2 r dΓe Ie7 = ∫ Γe pJm (χmn a r ) r dΓe− (P II+ mn +P II− mn ) ∫ Γe Jm (χmn a r )2 r dΓe (3.11) where Ωe and Γe are, respectively, the elementary triangular and two-node finite elements. 3.4.1. Elementary computation of the triangular finite element For the triangular finite element composed of three nodes, the integral Ie1 is written as follows Ie1 = [q1,q2,q3](Ke)1[p1,p2,p3] T (Ke)1 = ∫ Ωref −(∇q ·∇pT)detjr dξdη + ∫ Ωref ( iω c      N ′1 N ′2 N ′3      +U0 ·∇q ) ( − iω c [N ′1,N ′ 2,N ′ 3]+U0 ·∇p ) detjr dξdη (3.12) where pi = 1,2,3 and qi = 1,2,3 are, respectively, nodal acoustic pressures and nodal test functions of the triangular finite element. j is the inversematrix of the Jacobianmatrix J of the transformation from the reference element to the real base and N ′1(ξ,η), N ′ 2(ξ,η) and N ′ 3(ξ,η) are the interpolation functions of the triangular element (Dhatt and Touzout, 1989) N ′1(ξ,η) = 1− ξ−η N ′ 2(ξ,η) = ξ N ′ 3(ξ,η) = η (3.13) The integration of integral (3.12)2 is made using the numerical Gauss integration method, see Dhatt and Touzout (1989). Finally, the global correspondingmatrix is K1 = NelT ∑ 1 (Ke)1 (3.14) where NelT is the number of triangular finite elements. 318 M. Taktak et al. 3.4.2. Elementary computations of the two node finite element For the two-node finite element belonging to the lined part of the duct composed of two nodes, Ie2 and Ie3 are computed as follows Ie2 = [q1,q2](Ke)2 { p1 p2 } (Ke)2 =(Ke)21+(Ke)22+(Ke)23 (Ke)21 = ρ0iω 1 ∫ −1 { N1 N2 } [N1,N2] [N1,N2] [Z1,Z2] { N1 N2 } Le 2 r dξ (Ke)22 =−2ρ0U0 1 ∫ −1 { N1 N2 }        2 Le [−1/2,1/2] [Z1,Z2] { N1 N2 } − [N1,N2] 2 Le [Z1,Z2] { −1/2 1/2 } ( [Z1,Z2] { N1 N2 })2        Le 2 r dξ (Ke)23 = ρ0U 2 0 iω 1 ∫ −1 2 Le { −1/2 1/2 }        2 Le [−1/2,1/2] [Z1,Z2] { N1 N2 } − [N1,N2] 2 Le [Z1,Z2] { −1/2 1/2 } ( [Z1,Z2] { N1 N2 })2        Le 2 r dξ (3.15) where pi =1,2 and qi =1,2 are, respectively, nodal acoustic pressures and nodal test functions of the two-node finite element. Z1 and Z2 are the acoustic impedance of each node of the two-node finite element. Le is the finite element length, N1(ξ) and N2(ξ) are the interpolation functions of the two-node finite element defined by Dhatt and Touzout (1989) N1(ξ,η) = 1− ξ 2 N2(ξ)= 1+ ξ 2 (3.16) The computation of Ie3 is done for the two-node finite elements on the lined part extremities in which the first node of the first finite element of this part and the second node of the last finite element of the lined part are used Ie3 = [q1,q2](Ke)3Z2 { p1 p2 } − [q1,q2](Ke)3Z1 { p1 p2 } (Ke)3Z2 = ρ0U 2 0 iω 2 Le { 0 1 }        [−1/2,1/2] [Z1,Z2] { 0 1 } − [0,1] [Z1,Z2] { −1/2 1/2 } ( [Z1,Z2] { 0 1 })2        [r1,r2] { 0 1 } (Ke)3Z1 = ρ0U 2 0 iω 2 Le { 1 0 }        [−1/2,1/2] [Z1,Z2] { 1 0 } − [1,0] [Z1,Z2] { −1/2 1/2 } ( [Z1,Z2] { 1 0 })2        [r1,r2] { 1 0 } (3.17) where r1 and r2 are the radii of each corresponding real node. The integration of the above integrals ismade using the numerical Gauss integrationmethod, see Dhatt andTouzout (1989). The assembly of different elementary integrals computed before is obtained as follows K2,3 = NelLD ∑ 1 (Ke)3+(Ke)3Z1+(Ke)3Z2 (3.18) where NelLD is the number of two node finite elements along the lined part. Numerical characterization of an axisymmetric lined duct... 319 The integral Ie6 is written as follows for a finite element belonging to the left boundary Ie4 = [q1,q2](Ke) + 4 (P I+ mn)Nr +[q1,q2](Ke) − 4 (P I− mn)Nr (Ke) ± 4 =      · · · [−ik±mn(1+M 2 0)−kM0] 1 ∫ −1 N1(ξ)Jm ( χmn a r ) Le 2 r dξ · · · · · · [−ik±mn(1+M 2 0)−kM0] 1 ∫ −1 N2(ξ)Jm ( χmn a r ) Le 2 r dξ · · ·      2Nr (3.19) The integral Ie5 is written as follows for an two-node finite element belonging to the right boundary Ie5 = [q1,q2](Ke) + 5 (P II+ mn )Nr +[q1,q2](Ke) − 5 (P II− mn )Nr (Ke) ± 5 =      · · · [ik±mn(1+M 2 0)−kM0] 1 ∫ −1 N1(ξ)Jm ( χmn a r ) Le 2 r dξ · · · · · · [ik±mn(1+M 2 0)−kM0] 1 ∫ −1 N2(ξ)Jm ( χmn a r ) Le 2 r dξ · · ·      2Nr (3.20) By using linear interpolation of the pressure, the integrals Ie6 and Ie7 are obtained as follows Ie6 =(Ke)61 { p1 p2 } +(Ke) + 62(P I+ mn)Nr +(Ke) − 62(P I− mn)Nr Ie7 =(Ke)71 { p1 p2 } +(Ke) + 72(P II+ mn )Nr +(Ke) − 72(P II− mn )Nr (Ke)61 =(Ke)71 =        ... ... 1 ∫ −1 N1(ξ)Jm ( χmn a r ) Le 2 r dξ 1 ∫ −1 N2(ξ)Jm ( χmn a r ) Le 2 r dξ ... ...        2Nr (Ke) + 62 =(Ke) − 62 =(Ke) + 72 =(Ke) − 72 =  diag ( 1 ∫ −1 Jm (χm a r )2Le 2 r dξ )   Nr×Nr (3.21) Once the elementary integrals are computed, the assembly of them is obtained as follows K ± 4 = NelL ∑ 1 (Ke) ± 4 K ± 5 = NelR ∑ 1 (Ke) ± 5 (3.22) where NelL and NelR are, respectively, the number of two-node elements at the left and right boundaries K61 = NelL ∑ 1 (Ke)61 K ± 62 = NelL ∑ 1 (K±e )61 K71 = NelR ∑ 1 (Ke)71 K ± 72 = NelR ∑ 1 (K±e )72 (3.23) 320 M. Taktak et al. The arrangement of the previous system leads to the following matrix system        KM×M (K − 4 )M×Nr (K + 4 )M×Nr (K − 5 )M×Nr (K + 5 )M×Nr (K61)Nr×M (K − 62)Nr×Nr (K + 62)Nr×Nr 0 0 0 0 0 0 0 0 0 0 0 0 (K71)Nr×M 0 0 (K − 72)Nr×Nr (K + 72)Nr×Nr                                      p1 ... pM      M (PI−mn)Nr (PI+mn)Nr (PII−mn )Nr (PII+mn )Nr                          =0 (3.24) KM×M =K1+K2,3 with M is the number of nodes. For a given m, the azimuthal scattering matrix is defined as { PI−mn PII+mn } = s2Nr×2Nr { PI+mn PII−mn } (3.25) This matrix is obtained by formulating the system of Eq. (3.24)1 as follows Kp+A { PI+mn PII−mn } +B { PI−mn PII+mn } =0 Cp+U { PI+mn PII−mn } +V { PI−mn PII+mn } =0 (3.26) where p is the nodal acoustic pressure vector, and thematrices A,B,C,U, and V are defined as A= [ K − 4K + 5 ] B= [ K + 4K − 5 ] C=K61+K71 U= [ K − 62K + 72 ] V= [ K + 62K − 72 ] (3.27) The azimuthal scattering matrix is then written as s=(V−CK−1B−1)(U−CK−1A−1) (3.28) The total scatteringmatrix of the studied duct S2N×2N is obtained by repeating this operation for each m and by gathering the azimuthal matrices s2Nr×2Nr and N is the total number of modes present in the duct. 4. Computation of the acoustic power attenuation The axial acoustic intensity at a point M(r,θ,z) located in a plane section of the duct is given by Ville and Foucart (2003) Iz(r,θ,z) = 1 2 (1+M20)Re(P,V ∗ z )+ ρ0,V0 2 Re(VzV ∗ z )+ V0 2ρ0c 2 0 (PP∗) (4.1) where Vz is the axial acoustic velocity and P is the acoustic pressure. The acoustic power is given by W(z)= +∞ ∑ m=−∞ ∞ ∑ n=0 Iz,mn(z)Nmn (4.2) with Nmn is the normalization coefficient associated with the (m,n) mode defined as Nmn =SJ 2 m(χmn) ( 1− m2 χ2mn ) (4.3) where S=πa2 is the plane section are of the duct. Numerical characterization of an axisymmetric lined duct... 321 The axial acoustic intensity associated with the (m,n)mode Iz,mn is given by the following expression in function of modal acoustic pressures and velocities Iz,mn(z)= 1 2 (1+M20)Re(PmnV ∗ z,mn)+ ρ0V0 2 Re(Vz,mnV ∗ z,mn)+ V0 2ρ0c 2 0 Re(PmnP ∗ mn) (4.4) From this expression, the incident, reflected, transmitted and retrograde modal intensities are given by II+z,mn = (1+M20)Nmnk + mn 2ρ0c0(k−M0k + mn) |PI+mn| 2 II−z,mn = (1+M20)Nmnk − mn 2ρ0c0(k−M0k − mn) |PI−mn| 2 III+z,mn = (1+M20)Nmnk + mn 2ρ0c0(k−M0k + mn) |PII+mn | 2 III−z,mn = (1+M20)Nmnk − mn 2ρ0c0(k−M0k − mn) |PII−mn | 2 (4.5) The acoustic power attenuation Watt of a two-port duct is defined as the ratio between the acoustic power of incoming pressures from the two sides of the duct W in and the acoustic power of out-coming pressures from the two sides of the duct Wout Watt(dB)= 10log W in Wout W in = P ∑ m=−P Q ∑ n=0 (1+M20)Nmn 2ρ0c0 ( k+mn k−M0k + mn |PI+mn| 2+ k−mn k−M0k − mn |PII−mn | 2 ) Wout = P ∑ m=−P Q ∑ n=0 (1+M20)Nmn 2ρ0c0 ( k−mn k−M0k − mn |PI−mn| 2+ k+mn k−M0k + mn |PII+mn | 2) (4.6) The acoustic power attenuation is then written as follows Watt(dB)= 10log W in Wout =10log ∑2N i=1 |di| 2 ∑2N i=1λi|di| 2 (4.7) where λi are the eigenvalues of H defined as H2N×2N = [ [diag(XO)]2N×2NS2N×2N[diag(XI)] −1 2N×2N ]T∗ 2N×2N · [ [diag(XO)]2N×2NS2N×2N[diag(XI)] −1 2N×2N ] 2N×2N XImn = √ Nmn 2ρ0c0 ((1+M20)k + mn k−M0k + mn + k+mnM0 (k−M0k + mn)2 +M0 ) XOmn = √ Nmn 2ρ0c0 ((1+M20)k − mn k−M0k − mn + k−mnM0 (k−M0k − mn)2 +M0 ) d2N =U T∗ 2N×2N(Π in)2N (4.8) with U is the eigenvector matrix of H and T∗ denotes conjugate transpose. 5. Numerical results 5.1. Scattering matrix coefficients The studied duct in this paper is a 1 meter long cylindrical duct composed of three parts: 0.35m rigid wall duct, 0.3 lined duct and 0.35m rigid wall duct. This duct is similar to the experimental duct used by Taktak et al. (2010). The computation of the multimodal scattering 322 M. Taktak et al. matrix and the acoustic power attenuation is made by supposing that the duct is lined by a Helmholtz resonator composed of a perforated plate with the thickness e = 0.8mm, the hole diameter d = 1mm with a perforation ratio σ = 5% of the honey comb structure with thickness D = 20mm and a rigid wall plate. This kind of liner is characterized by its acoustic impedance Z. In the present work, the acoustic impedancemodel of Elnady and Boden (2003) is used as the input for computation of the numerical multimodal scattering matrix and the acoustic power attenuation of the studied duct. This model gives the resonance frequency at ka = 2.22. Computations are made for different Mach numbers (M0 = 0, 0.1, 0.2) over the frequency band ka∈ [0,3.8] to evaluate the flow effect. Fig. 2. Modulus of the transmission coefficients T+ 00,00 (a) and T+ 10,10 (b) versus ka for severalMach numbers Figures 2a,b present the moduli of transmission coefficients T+00,00 and T + 10,10 computed in the same direction of the flow versus of the nondimensional wave number ka for different Mach numbers. The modulus of the coefficient T+00,00 shows that it is near 1 in ka ∈ [0,0.8]. From ka = 0.8, this modulus decreases with the frequency until becoming nil in the interval ka ∈ [2.4,2.8] near the theoretical resonance frequency. Then, an increase of the modulus is observed in the rest of the studied frequency band until reaching 0.4 at ka = 3.8. For the T+10,10 modulus, an increase versus ka is observed from ka = 2.8 to reach 0.4 at ka = 3.8. Figures 2a,b also showthat there are no significant effects of theflowon transmission coefficients. Figures 3a,b,c present, respectively, themoduli of reflection coefficients R+00,00,R + 10,10 and R + 20,20 Fig. 3. Modulus of the reflection coefficients R+ 00,00 (a),R+ 10,10 (b) and R+ 20,20 (c) versus ka for several Mach numbers Numerical characterization of an axisymmetric lined duct... 323 of the studied duct. Oscillations are observed on the reflection coefficient R+00,00. The reflection coefficients of higher order modes are close to 1 near the cut on frequencies, then a decrease of theirmoduli is observed versus ka. Figures 3a,b,c show the flow effects on reflection coefficients: when the flow velocity increases, the reflection coefficients moduli decrease except the R+00,00 coefficientmodulus in ka∈ [1.2,1.8].Thisdecrease ismoreapparentonthe (0,0)mode reflection coefficient (∼ 0.05) and (2,0) mode (∼ 0.2) and less important than the (1,0) mode. 5.2. Acoustic power attenuation Acoustic power attenuations are computed using a configuration of unit modal incident pressures from one side of the duct (left) and in the same direction of flow, see Taktak et al. (2010) (Pin = [1,1,1,1,1,0,0,0,0,0]T). Figures 4a,b,c present the acoustic power attenuation of the studied duct versus ka, respectively, in presence of (0,0), (1,0), (2,0) for different studied Mach numbers. They show that attenuations are dependent of the incident wave and that the maximum of attenuation is observed near the liner resonance frequency. The amplitude and the frequency of thismaximum is dependent on the flow speed. Figure 4a shows that thismaximum is equal to 15dB without flow at ka = 3.1, 17dB for M0 = 0.1 at ka = 3 and 19dB for M0 =0.2 at ka=2.95. The same remark is observed in presence of the (1,0) and (2,0) mode: withoutflow, themaximumof attenuation inpresenceof the (1,0) is 12dBat ka=3.2, 13dBat ka=3.1 for M0 =0.1 and 14dB at ka=3.05 for M0 =0.2. These figures allowed concluding that an increase in the flow velocity generates a increase in the acoustic power attenuation and a decrease in the maximum of attenuation frequency. Fig. 4. Acoustic power attenuation of the studied duct in the presence of (0,0) mode (a), (1,0) mode (b) and (2,0) mode (c) versus ka for severalMach numbers 6. Conclusions In this study, a numerical method for the characterization of a lined duct inthe presence of flowwas developed and presented. Thismethod is based on the computation of themultimodal scattering matrix as well as the acoustic power attenuation. By varying the flow velocity, its effect was evaluated: the increase of the flow decreases the reflection coefficients when the effect 324 M. Taktak et al. is weak on the transmission coefficients. For the acoustic power attenuation, the increase of the flowvelocity increases the attenuation anddecreases the frequency of themaximumattenuation. Acknowledgements This work was carried out in the framework of Tunisian-French research project DGRSRT/CNRS 09/R 11-43 on the modeling of the vibro-acoustic problems. References 1. AbomM., 1991,Measurementof the scatteringmatrix of acoustical two-ports,Mechanical Systems Signal Processing, 5, 2, 89-104 2. Akoum M., Ville J.M., 1998, Measurement of reflection matrix of a discontinuity in a duct, Journal of the Acoustical Society of America, 103, 5, 2463-2468 3. Aurégan Y., Starobinski R., 1998, Determination of acoustical energy dissipation/production potentiality fromthe acoustic transfer functions of amultiport,ActaAcusticaUnited withAcustica, 85, 788-792 4. Bi W.P., Pagneux V., Lafarge D., Aurégan Y., 2006, Modelling of sound propagation in non-uniform linedduct using aMulti-ModalPropagationMethod, Journal of Sound andVibration, 289, 1091-1111 5. 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TaktakM., Ville J.M., HaddarM.,GabardG., Foucart F., 2010,An indirectmethod for the characterization of locally reacting liners, Journal of the Acoustical Society of America, 127, 6, 3548-3559 Numerical characterization of an axisymmetric lined duct... 325 17. To C.W.S., Doige A.G., 1979, A transient testing technique for the determination of matrix parameters of acoustic systems, I: Theory and principles, Journal of Sound and Vibration, 62, 207-222 18. To C.W. S., Doige A.G., 1979, A transient testing technique for the determination of matrix parameters of acoustic systems, II: Experimental procedures and results, Journal of Sound and Vibration, 62, 223-233 19. Ville J.M., Foucart F., 2003, Experimental set up for measurement of acoustic power dissipa- tion in lined ducts for higher order modes propagation with air mean flow conditions, Journal of the Acoustical Society of America, 114, 4, 1742-1748 Numeryczna charakteryzacja wyściełanego przewodu osiowo-symetrycznego z przepływem za pomocą wielomodalnej macierzy rozpraszania Streszczenie W pracy zaprezentowano numeryczną metodę wyznaczania macierzy rozpraszania dla wyściełanego przewoduzuwzględnieniemwewnętrznegoprzepływuczynnika.Metodę opartona zastosowaniu równania konwekcji Helmholtza z wprowadzeniem ciśnień modalnych na brzegach jako dodatkowych stopni swo- body układu. Efekty brzegowe na wlocie i wylocie przewodu falowego o skończonej długości pominięto. Wybórmacierzy rozpraszania uzasadniono faktem, że reprezentuje onawewnętrzną charakterystykę ana- lizowanegomodelu. Zaproponowanyelement skończony zweryfikowanopoprzez porównanie z istniejącymi rozwiązaniamianalitycznymidla prostychprzypadkówkonfiguracji przewodu.Następnie numerycznie ob- liczono wartości elementów macierzy rozpraszania oraz współczynniki tłumienia akustycznego dla kilku prędkości przepływuw celu określenia, jak dalece wpływa on na badany układ. Manuscript received March 9, 2012; accepted for print June 14, 2012