10 ` eme Congr ` es Fran¸ cais d’Acoustique Lyon, 12-16 Avril 2010 A mixed analytical-numerical model for the S-matrix computation of bidimensional lined ducts with HQ tubes Romain Marechal 1 , Emmanuel Perrey-Debain 1 , Jean-Michel Ville 1 , Benoˆ ıt Nennig 1 1 UMR CNRS 6253 Laboratoire Roberval, Rue Personne de Roberval, F-60200 Compi` egne, {romain.marechal,emmanuel.perrey-debain,jean-michel.ville,benoit.nennig}@utc.fr The Herschel-Quincke (HQ) tubes, consisting in putting tubes in derivation along a main acoustic wave guide, are used as passive devices to control fan noise. In order to assess the efficiency of this system, a new mixed analytical-numerical model is presented. The technique relies on combining Finite Element techniques to accurately describe the HQ tube with an integral representation for the acoustic pressure in the main duct. The presence of acoustic liners on the walls of the duct is taken into account via an appropriate modal decomposition of the Green’s function. We show that our algorithm allows a very fast and accurate computation of the scattering matrix of such a system with a numerical complexity that grows very mildly with the frequency. Results show that ‘nearly’ optimal configurations can be quickly identified with a very small computational expense. 1 Introduction One of the most significant sources of noise of an air- craft is due to the propelling system. This noise, which is present during all phases flight around airport, can be decomposed into several types : classical jet noise out- side the exhaust nozzle and inner turbo machinery noise (fan, compressor, turbine & combustion). In particular, fan noise is responsible for pure tones at the Blade Pas- sage Frequency (BPF) harmonics, due to the interaction between the rotor wakes and the stator vanes. In order to reduce noise level in modern turbofan engines, sound waves generated by the fan are typically absorbed by acoustic lining covering the duct engine. Though effi- cient, these passive liners seem to have reach their limit and there is still a need for considering other passive techniques to reduce further the sound radiation from the duct outlet. In this context Herschel-Quincke tubes concept could prove to be a reliable option. In 1833, Herschel [8] first discussed the idea of us- ing acoustic interferences of tones by simply connect- ing a tube to the main duct in view of reducing the transmitted acoustic waves. Thirty three years later, Quincke [14] experimentally validated Herschel’s theory and many works and experiments have been carried out to explain physical phenomena and explore the poten- tiality of this system as a noise control device [3, 18]. The assessment of the efficiency of such a system re- quires a precise knowledge of the acoustic field in the duct. Though standard Finite Element (FE) software could, in principle, be used for this purpose, a full 3D FE model would be extremely demanding as the num- ber of variables is expected to grow like f 3 (f is the frequency). This can have a negative impact when, for instance, some efficient optimizations (geometry of the HQ tubes and their positions) are needed. Assuming plane wave propagation, the resonance be- havior of two duct combination was first established analytically by Selamet et al. [16] and then extended to a multiple duct configuration [17]. The proposed approach is simple to implement and allows a very fast computation of the transmitted wave but it is unfortunately limited to low frequency applications. To make some progress, Brady [2] proposed a two- dimensional model including multi-modal analysis in the main duct using a Green’s function formalism. The three-dimensional model was then extended by Hallez [6]. Finally, Poirier [12, 13] proposed an improvement by taking into account the exact shape of the interface between the main cylindrical duct and the HQ tube. All the authors just cited simplified their analysis by assuming that the acoustic velocity is constant over the duct-tube interface. Furthermore they all modelized the HQ tubes as if they were straight waveguides in which only plane waves are allowed to propagate. Because these assumptions are known to break down as the frequency increases (see for instance Tang & Lam [19]), Mar´ echal et al. [10] proposed an enhanced model by taking into account (i) the exact shape of the HQ tube(s) and (ii) the non uniformity of the acoustic ve- locity on the interfaces. Authors show that these im- provements can be made with a relatively small addi- tional computational cost while leading to very accu- rate results even in the mid-frequency regime. In the present paper, we shall show that the technique can be also extended and applied for ducts with acoustic lin- ers on its walls. This is particularly important as early experiments combining typical acoustic liners and HQ tubes already showed promising results for reducing the transmitted acoustic power [4, 1].