Computational Aeroacoustics

Computational Aeroacoustics Duct Acoustics Gwénaël Gabard Institute of Sound and Vibration Research University of Southampton, UK gabard@soton.ac.uk ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 1 / 36

Introduction Review basic properties of duct modes Sound sources in ducted flows Modal analysis for more complex configurations CAA techniques ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 2 / 36

Duct modes ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 3 / 36

Definition of duct modes Consider a straight duct with axis x and an arbitrary cross section S. The duct contains a uniform mean flow u 0 and the walls are rigid. The sound field satisfies the convected wave equation 1 c 2 0 D 2 0p Dt 2 2 p = 0, (1) S y z C x where c 0 is the speed of sound and D 0/Dt = / t + u 0 / x is the material derivative. We seek solution of the form (normal modes) p = Ψ(y, z)e ikx x+iωt, (2) where k x is the axial wavenumber and Ψ is the mode shape function. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 4 / 36

Uniform flow with hard walls The mode shape functions Ψ m are solutions of the following eigenvalue problem defined on the cross section: k Ψ 2 + 2 Ψ = 0 on S, (3) Ψ = 0 along C. (4) n And the axial wavenumber is linked to the transverse wavenumber k by the dispersion relation: k ± x = k 0 ±η m M 1 M 2, (5) where k 0 = ω/c 0, M = u 0/c 0 is the flow Mach number and Im(k x /k 0 ) 6 4 2 0 2 4 6 4 2 0 2 4 Re(k x /k 0 ) η m = 1 (1 M 2 )k 2 /k2 0 (6) is the cut-on ratio of the mode. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 5 / 36

Uniform flow with hard walls Propagating (cut-on) or evanescent (cut-off) modes: If k x is imaginary: the mode decays exponentially and is evanescent. It represents near-field effects close to the source. In isolation a cut-off mode does not contribute to the acoustic power. If k x is real: the mode is propagating and its amplitude remains constant along the duct. The mean flow reduces the cut-off frequencies. Orthogonality: the modes form an orthogonal basis: Ψ mψ n ds = N mδ mn. (7) S Modal basis: any sound field in the duct can be described as a linear combination of the normal modes: p = (A +me ) ik+ xm x + A m e ik xm x Ψ m(y, z). (8) m=1 The modal amplitudes A ± m are defined by the source of sound. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 6 / 36

Uniform flow with hard walls Circular duct: the mode shape functions are Ψ = J m(k r r)e imθ, (9) where J m is the Bessel function and m is the azimuthal order. m = 0: axi-symmetric modes: no θ dependence. m 0: spinning modes: the angular velocity of the mode is given by ω/m: [ p exp im (θ ω )] m t. (10) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 7 / 36

Uniform flow and lined walls Myers condition: a surface with impedance Z(ω) under an infinitely thin boundary layer (Myers, 1980; Ingard, 1959). The eigenvalue problem becomes: y boundary layer y k Ψ 2 + 2 Ψ = 0 on S, (11) Ψ ρ0c0 = i (k 0 Mk x) 2 Ψ. n Zk 0 (12) u 0 (y) The mode shape functions are different between the left and right modes, so the modal expansion now reads: p = A + mψ + m(y, z)e ik+ xm x + A m Ψ m (y, z)e ik xm x. (13) m=1 ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 8 / 36

Uniform flow and lined walls Surface waves: some modes are only present very close to the duct wall and are associated with the hydrodynamic oscillations in the thin boundary layer. The number of surface waves depends on the parameters (Rienstra, 2003a; Brambley and Peake, 2006). Stability: a surface wave can be unstable and grow exponentially as it propagates along the wall. No clear distinction between cut-on and cut-off modes. The modes are not orthogonal. Dissipation rate 8.69 Im(k x) in db/m. Im(k x /k 0 ) 6 4 2 0 2 4 6 axial wavenumber (hard wall, lined) 2 0 2 4 6 8 Re(k x /k 0 ) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 9 / 36

Non-uniform flows Axial parallel sheared flow: The mean flow is independent of axial position. The perturbations are governed by the Pridmore-Brown (1958) equation. boundary layer mean velocity profile [ ] D 0 1 D 2 0p Dt c0 2 Dt 2 2 p + 1 ( ( ρ 0) ( p) + 2( u 0) p ) = 0. (14) ρ 0 x It describes both the acoustic and hydrodynamic modes (including the potential instability of the mean flow profile). One can include explicitly the boundary layer profiles, and use the standard impedance condition at the wall (v n = p/z). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 10 / 36

Non-uniform flows Swirling flows: Azimuthal and axial velocity components, representative of the interstage between the fan and the guide vanes on a turbofan engine. The mean swirl couples acoustic and hydrodynamic waves. Golubev and Atassi (1998), Tam and Auriault (1998) and Cooper and Crighton (2000). Different types of modes: Acoustic modes: pressure dominated sonic modes. Nearly-convected modes: vorticity dominated. Continuous spectrum associated with the critical layer (Heaton and Peake, 2006). No general definition of the acoustic energy: special cases such as high frequencies (Atassi, 2003). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 11 / 36

Practical aspects Uniform flows and canonical geometries: Scalar characteristics equation for k x, for instance for a lined circular duct: ρ 0c 0 (k 0 Mk x) 2 k r J m(k r R) = 1. (15) izk 0 J m(k r R) Standard zero-finding or minimization algorithms can be used. With liner, it is not trivial to ensure that all the modes are found. One technique is to start from the hard-wall case and then track the values of k x as the impedance is varied from infinity to the target impedance. See Eversman (1995) and Rienstra (2003a). Im(k x /k 0 ) 6 4 2 0 2 4 6 2 0 2 4 6 8 Re(k /k ) x 0 ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 12 / 36

Practical aspects With non-uniform flows: numerical solution of Pridmore-Brown s equation: Shooting method (Mungur and Plumblee, 1969; Eversman, 1971) Start with an initial guess for k x Integrate Pridmore-Brown s equation from one wall to the other. Check if the boundary condition at the wall is satisfied. Adjust k x and repeat until the boundary condition is satisfied. Finite difference schemes or finite elements (Moinier and Giles, 2005; Vilenski and Rienstra, 2007). Construct an algebraic eigenvalue problem of the form: Ap = k x Bp. Sort the acoustic modes from the hydrodynamic modes, based on heuristic rules (pressure dominated, vorticity dominated, etc). Non canonical geometries: Finite element solution of the eigenvalue problem (Gabard and Astley, 2008). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 13 / 36

Scattering matrix A passive component in a duct system (e.g. air conditioning systems) can be described in terms of scattering matrix in the frequency domain: ( ) [ ] ( ) A RA T B + = BA A + T AB R B B. (16) A + A B + x B Transmission matrices T AB and T BA. Reflection matrices R A and R B. These matrices can be obtained using theoretical modelling, measurements, or CAA predictions. Commonly used for plane wave propagation (2-port and multi-port components). See Munjal (1987), Davies (1988) and Åbom (1991). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 14 / 36

Sources in ducts ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 15 / 36

Sources in ducts If we consider a source radiating inside a duct, we can use the modal expansion of the sound field to obtain the solution. For instance for a monopole at x s in a hard-wall duct with uniform flow we can write ( ) 1 D 2 0 Dt 2 2 G ω(x x s) = s(ω)δ(x x s), (17) c 2 0 The solution can be written as a modal expansion: Ψm(ys, zs) G ω(x x s) = s(ω) Ψ m(y, z)e ik ± xm (x xs ), for x x s, (18) 2iN mη mk 0 m where the amplitude of each mode can be easily identified, and the coefficients N m are the norms of the mode shape functions N m = For lined ducts see Rienstra and Tester (2008). Ψ m 2 ds. (19) S ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 16 / 36

Sources in ducts Behaviour near cut-off frequencies: a mass point source in a circular duct with uniform flow (M = 0.5) 40 20 p 2 [db] 0 20 40 0 2 4 6 8 10 12 Helmholtz number 20 Power [db] 0 20 40 0 2 4 6 8 10 12 Helmholtz number ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 17 / 36

Sources in ducts The acoustic response of the duct acts as a filter due to the cut-off frequencies of the modes. Convective amplification and Doppler effects due to source motion are also present. Near the cut-off frequency of a mode: The system is very sensitive to small changes in parameters or to small amounts of numerical error. High amplitude oscillations might cause problems for numerical simulations. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 18 / 36

Tones vs broadband noise For example, for fan noise we generally distinguish between: Single mode tones (deterministic) Multi-mode tones (deterministic) Broadband noise: random (generated by turbulence) and also multi-mode. Generally defined using a modal correlation matrix: C mn(ω) = A m(ω)a n(ω). An assumption commonly used for broadband noise is that of uncorrelated modes with equal energy per mode (Joseph et al., 2003). from Aeroacoustics of Flight Vehicles, Ed. by H. Hubbard, p. 154. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 19 / 36

Multi-port approach Active duct component can also be considered: ( ) [ ] ( ) ( ) A RA T B + = BA A + S T AB R B B + S +. (20) A + source B + x The modes S and S + generated by the source inside the duct segment. A B Difficulty to separate transmission line into individual components, due to feedback mechanisms between them. Self-sustained oscillations (whistling) can appear. The source terms depends on the incident sound field (Karlsson and Åbom, 2011). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 20 / 36

Non unifom impedance ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 21 / 36

Non-uniform impedance Circumferentially varying impedance: Multimodal methods: combine modes from the fully lined and hard-wall configurations (Bi et al., 2006; Bi et al., 2007) Fourier decomposition (Fuller, 1984; Tam et al., 2008). Cargill method for small splices using a Kirchhoff integral (Tester et al., 2006). Finite elements (Gabard and Astley, 2008). z liner y ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 22 / 36

Non-uniform impedance Axially segmented liners: Matching two different modal basis The impedance discontinuity scatters the incident modes A + and B into the modes A and B +. Matching techniques: Pressure and axial velocity. Fluxes of mass and axial momentum (Gabard and Astley, 2008). A + A segment 1 segment 2 matching plane B + x B Without flow these techniques are equivalent. With flow the solutions will differ, especially the behaviour of the sound field at the discontinuity. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 23 / 36

Curved ducts Multiple-scales methods for slowly varying ducts (Nayfeh, 1973; Nayfeh et al., 1975; Rienstra, 1999; Rienstra, 2003b) Multimodal approach with no flow (Félix and Pagneux, 2001; Félix and Pagneux, 2004). See also Brambley and Peake (2008). from Rienstra & Eversman, JFM 437, 2001. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 24 / 36

CAA techniques ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 25 / 36

CAA techniques Physical models: Linearized Euler equations, Linearized potential theory, Acoustic perturbation equations. broadband noise radiating from annular duct Numerical schemes: Time domain or frequency domain. Finite difference, finite volume, finite elements, discontinuous Galerkin method. Use of modal basis as boundary condition to define incoming sound field and act as non-reflecting conditions. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 26 / 36

CAA techniques Methods commonly used: Linearized potential theory solved in the frequency domain using finite elements (Astley et al., 2011). Linearized Euler equations solved in the time domain using finite difference schemes (Özyörük et al., 2004; Zhang et al., 2005). Linearized Euler equations solved in the time domain using discontinuous Galerkin methods (Atkins and Lockard, 1999). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 27 / 36

Time-domain impedance y y y y Simulating acoustic liner with flow in the time domain presents specific issues. 1 0.8 (a) Interaction with the boundary layer: One can resolve completely the boundary layer but this is costly, or use Myers condition (infinitely thin boundary layer) It is ill-posed in the time domain (Brambley, 2009). Numerical instabilities can be triggered depending on the accuracy and dissipation of the numerical scheme (Gabard and Brambley, 2014). Alternatives have been proposed where a small but finite boundary layer thickness is used (Rienstra and Darau, 2011; Brambley, 2011). 0.6 0.4 0.2 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 1 (b) 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 1 0.8 0.6 (c) 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 1 (d) 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 28 / 36

Time-domain impedance Implementation of the impedance in the time domain: When converted to the time domain the expression Z(ω)v n(ω) = p(ω) corresponds to a convolution product between v n(t) and the impulse response function z(t) of the liner. Various constraints have to be satisfied to obtain a meaningful and stable implementation (causality, stability, etc) (Rienstra, 2006). Different implementations have been proposed for the convolution product (Tam and Auriault, 1996; Özyörük et al., 1998) Use of strong damping or filtering to obtain a stable solution, often adjusted case by case. Comparison of CAA predictions against NASA GIT experimental data (Richter et al., 2011) 1kHz 2kHz ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 29 / 36

Scattering matrix CAA methods can be used to predict the scattering matrices and the sources generated within duct components such as fan, bifurcations, etc. Large Eddy Simulations (Lacombe et al., 2013). Linearised Navier-Stokes equations to predict scattering matrix of a two-port component (Kierkegaard et al., 2010; Kierkegaard et al., 2012a; Kierkegaard et al., 2012b) Plane wave propagating through an in-duct orifice. Note the vorticity shed from the sharp edge. From Kierkegaard et al. (2012b). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 30 / 36

Further reading The aeroacoustics of ducted flows is discussed in Chapter 13 and 14 in the book edited by Hubbard (1991), by Rienstra and Hirschberg (2014) and Astley (2009). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 31 / 36

References I Åbom, M. (1991). Measurement of the scattering-matrix of acoustical two-ports. In: Mechanical Systems and Signal Processing 5.2, pp. 89 104. Astley, R. (2009). Numerical methods for noise propagation in moving flows, with application to turbofan engines. In: Acoustical Science and Technology 30.4, pp. 227 239. Astley, R., R. Sugimoto and P. Mustafi (2011). Computational aero-acoustics for fan duct propagation and radiation. Current status and application to turbofan liner optimisation. In: Journal of Sound and Vibration 330.16, pp. 3832 3845. Atassi, O. V. (2003). Computing the sound power in non-uniform flow. In: Journal of sound and vibration 266.1, pp. 75 92. Atkins, H. L. and D. P. Lockard (1999). A high-order method using unstructured grids for the aeroacoustic analysis of realistic aircraft configurations. In: AIAA paper 1945, p. 1999. Bi, W. P. et al. (2006). Modelling of sound propagation in a non-uniform lined duct using a multi-modal propagation method. In: Journal of sound and vibration 289.4, pp. 1091 1111. Bi, W. et al. (2007). An improved multimodal method for sound propagation in nonuniform lined ducts. In: The Journal of the Acoustical Society of America 122.1, pp. 280 290. Brambley, E. (2009). Fundamental problems with the model of uniform flow over acoustic linings. In: Journal of Sound and Vibration 322, pp. 1026 1037. (2011). Well-posed boundary condition for acoustic liners in straight ducts with flow. In: AIAA Journal 49.6, pp. 1272 1282. Brambley, E. and N. Peake (2006). Classification of aeroacoustically relevant surface modes in cylindrical lined ducts. In: Wave Motion 43, pp. 301 310. Brambley, E. and N Peake (2008). Sound transmission in strongly curved slowly varying cylindrical ducts with flow. In: Journal of Fluid Mechanics 596, pp. 387 412. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 32 / 36

References II Cooper, A. and D. Crighton (2000). Global modes and superdirective acoustic radiation in low-speed axisymetric jets. In: Eur. J. Mech. B - Fluids 19, pp. 559 574. Davies, P. (1988). Practical flow duct acoustics. In: Journal of Sound and Vibration 124.1, pp. 91 115. Eversman, W. (1971). Effect of boundary layer on the transmission and attenuation of sound in an acoustically treated circular duct. In: Journal of the Acoustical Society of America 49.5, pp. 1372 1380. (1995). Theoretical models for duct acoustic propagation and radiation. In: Aeroacoustics of flight vehicles Theory and practice. Ed. by H. Hubbard. Acoustical Society of America. Félix, S and V Pagneux (2001). Sound propagation in rigid bends: A multimodal approach. In: The Journal of the Acoustical Society of America 110.3, pp. 1329 1337. (2004). Sound attenuation in lined bends. In: The Journal of the Acoustical Society of America 116.4, pp. 1921 1931. Fuller, C. (1984). Propagation and radiation of sound from flanged circular ducts with circumferentially varying wall admittances, II: finite ducts with sources. In: Journal of Sound and Vibration 93.3, pp. 341 351. Gabard, G. and R. Astley (2008). A computational mode-matching approach for sound propagation in three-dimensional ducts with flow. In: Journal of Sound and Vibration 315, pp. 1103 1124. Gabard, G. and E. Brambley (2014). A Full Discrete Dispersion Analysis of Time-Domain Simulations of Acoustic Liners with Flow. In: Journal of Computational Physics. In press. Golubev, V. and H. Atassi (1998). Acoustic-vorticity waves in swirling flows. In: Journal of Sound and Vibration 209.2, pp. 203 222. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 33 / 36

References III Heaton, C. and N. Peake (2006). Algebraic and exponential instability of inviscid swirling flow. In: Journal of Fluid Mechanics 565, pp. 279 318. Hubbard, H. H. (1991). Aeroacoustics of flight vehicles: Theory and practice. volume 2. noise control. Tech. rep. DTIC Document. Ingard, K. (1959). Influence of fluid motion past a plane boundary on sound reflection, absorption and transmission. In: Journal of the Acoustical Society of America 31.7, pp. 1035 1036. Joseph, P., C. Morfey and C. Lowis (2003). Multi-mode sound transmission in ducts with flow. In: Journal of Sound and Vibration 264, pp. 523 544. Karlsson, M. and M. Åbom (2011). On the use of linear acoustic multiports to predict whistling in confined flows. In: Acta Acustica united with Acustica 97.1, pp. 24 33. Kierkegaard, A., S. Boij and G. Efraimsson (2010). A frequency domain linearized Navier Stokes equations approach to acoustic propagation in flow ducts with sharp edges. In: The Journal of the Acoustical Society of America 127.2, pp. 710 719. (2012a). Simulations of the scattering of sound waves at a sudden area expansion. In: Journal of Sound and Vibration 331.5, pp. 1068 1083. Kierkegaard, A. et al. (2012b). Simulations of whistling and the whistling potentiality of an in-duct orifice with linear aeroacoustics. In: Journal of Sound and Vibration 331.5, pp. 1084 1096. Lacombe, R et al. (2013). Identification of aero-acoustic scattering matrices from large eddy simulation: Application to whistling orifices in duct. In: Journal of Sound and Vibration 332.20, pp. 5059 5067. Moinier, P. and M. B. Giles (2005). Eigenmode analysis for turbomachinery applications. In: Journal of propulsion and power 21.6, pp. 973 978. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 34 / 36

References IV Mungur, P. and H. Plumblee (1969). Propagation and attenuation of sound in a soft-walled annular duct containing a sheared flow. SP-207. NASA. Munjal, M. (1987). Acoustics of ducts and mufflers with application to exhaust and ventilation system design. Wiley-Interscience. Myers, M. (1980). On the acoustic boundary condition in the presence of flow. In: Journal of Sound and Vibration 71.3, pp. 429 434. Nayfeh, A. (1973). Perturbations methods. John Wiley & Sons. Nayfeh, A., J. Kaiser and D. Telionis (1975). Acoustics of aircraft engine-duct systems. In: AIAA Journal 13.2, pp. 130 153. Özyörük, Y., L. Long and M. Jones (1998). Time-domain numerical simulation of a flow-impedance tube. In: Journal of Computational Physics 146, pp. 29 57. Özyörük, Y et al. (2004). Frequency-domain prediction of turbofan noise radiation. In: Journal of Sound and Vibration 270.4, pp. 933 950. Pridmore-Brown, D. (1958). Sound propagation in a fluid flowing through an attenuating duct. In: Journal of Fluid Mechanics 4, pp. 393 406. Richter, C. et al. (2011). A review of time-domain impedance modelling and applications. In: Journal of Sound and Vibration 330.16, pp. 3859 3873. Rienstra, S. (1999). Sound transmission in slowly varying circular and annular lined ducts with flow. In: Journal of Fluid Mechanics 380, pp. 279 296. (2003a). A classification of duct modes based on surface waves. In: Wave Motion 37, pp. 119 135. Rienstra, S. (2003b). Sound propagation in slowly varying lined flow ducts of arbitrary cross-section. In: Journal of Fluid Mechanics 495, pp. 157 173. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 35 / 36

References V Rienstra, S. (2006). Impedance models in time domain, including the extended Helmholtz resonator model. In: 12th AIAA/CEAS Aeroacoustics Conference. AIAA paper 2006-2686. Cambridge, MA, USA. Rienstra, S. and M. Darau (2011). Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall. In: Journal of Fluid Mechanics 671, pp. 559 573. Rienstra, S. and A. Hirschberg (2014). An introduction to acoustics. Available at http://www.win.tue.nl/ sjoerdr/papers/boek.pdf. Rienstra, S. W. and B. J. Tester (2008). An analytic Green s function for a lined circular duct containing uniform mean flow. In: Journal of Sound and Vibration 317.3, pp. 994 1016. Tam, C. and L. Auriault (1996). Time-domain impedance boundary conditions for computational aeroacoustics. In: AIAA Journal 34.5, pp. 917 923. (1998). The waves modes in ducted swirling flows. In: Journal of Fluid Mechanics 371, pp. 1 20. Tam, C., H. Ju and E. Chien (2008). Scattering of Acoustic Duct Modes by Axial Liner Splices. In: Journal of Sound and Vibration 310, pp. 1014 1035. Tester, B. et al. (2006). Scattering of sound by liner splices: a kirchhoff model with numerical verification. In: AIAA journal 44.9, pp. 2009 2017. Vilenski, G. G. and S. W. Rienstra (2007). Numerical study of acoustic modes in ducted shear flow. In: Journal of Sound and Vibration 307.3, pp. 610 626. Zhang, X., X. Chen and C. Morfey (2005). Acoustic radiation from a semi-infinite duct with a subsonic jet. In: International Journal of Aeroacoustics 4, pp. 169 184. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Duct Acoustics 36 / 36