Guided convected acoustic wave coupled with a membrane wall used as noise reduction device

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1 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century PROCEEDINGS of the nd International Congress on Acoustics Structural Acoustics and Vibration (others): Paper ICA Guided convected acoustic wave coupled with a membrane wall used as noise reduction device Virgile Meyer (a), Vincent Martin (a) (a) Institut Jean Le Rond D'Alembert, UMR (CNRS/UPMC) 7190, 4 place Jussieu, 755 Paris Cedex 05, virgile.meyer@etu.upmc.fr, vincent.martin@upmc.fr Abstract It is known since long that an essentially plane guided acoustic wave coupled with a vibrating wall could be attenuated in a certain frequency range. Various types of yielding walls can be studied such as plates, membranes or shells. This paper will focus on an acoustic wave convected by a uniform flow, propagating in a duct and interacting with a membrane in a 3 dimensions configuration. The problem is solved in the domain with the yielding structure via a home-made finite element model and coupled with a pure analytical plane wave beyond. It will be shown numerically that attenuation with flow reduces the attenuation efficiency. As results have been published for a similar (but not totally identical) configuration for small Mach number (numerically with other methods, and experimentally) the comparison is possible. Keywords: Acoustic / structure coupling in ducts.

2 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century Guided convected acoustic wave coupled with a membrane wall used as noise reduction device 1 Introduction The guided acoustic propagation coupled with yielding wall results in waves of particular properties. Indeed, wavenumbers - real (propagated wave), imaginary (evanescent wave) or complex (attenuated wave) - may appear unique or multiple for a given frequency. For example, in the case of wall with local reaction, there is a frequency band called forbidden with evanescent waves resulting in a sound pressure reduction at the duct exit while outside the forbidden band, propagation is observed with sub- and supersonic waves which could also induce attenuation through impedance breaking [1, ]. Both of those phenomena lead to attenuation without any dissipation mechanism which could be used for industrial application in configuration implying ducts [3, 4]. With other kind of wall such as plate [5] or shell [6], attenuation and/or multiple branches for a frequency band are expected but are not always easily observed experimentally. The work presented here focuses on a model of a convected acoustic wave propagating in a 3D duct and coupled with a vibrating wall of the membrane-type. Two coupling conditions will be considered to take into account the flow and results will be presented from numerical and analytical simulations. It will be shown that the presence of flow reduces the attenuation. The considered configuration.1 Geometry Let us consider the following configuration as shown in Figure 1 : an infinite 3D waveguide with infinite rigid walls but on a part where the incident acoustic wave is coupled with a yielding wall (membrane model). The incident plane wave is also convected by a steady uniform flow. The largest width of the guide is 8 cm and therefore the first cut-off frequency appears at Hz in the parts where walls are rigid.

3 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century Figure 1: Infinite 3D guide with uniform flow and incident pressure coupled with a yielding wall (in yellow). The harmonic acoustic field is obtained by using the Helmholtz equation for the acoustic domain coupled with the membrane vibration equation.. Coupled convected acoustic-structure equations A steady and uniform flow is considered here and this convection appears in the model equation in two places: firstly in the convected acoustic equation which describes the wave propagation in the air medium and secondly in a coupling condition. The coupling between the acoustic domain and the yielding wall asks for the continuity of forcelike quantities and normal displacement at the interface. The continuity of force-like quantity is already written in the membrane equation (the acoustic pressure acts as an external force). The continuity of normal displacement rests on the time-domain dynamic equation with the normal n pointing outwards the guide and where ρ is the air density, p the acoustic pressure: p p d d n z dt dt x, y, L, t ρ w x, y, t ρ w x, y, t z s a (1) which is written firstly at the interface, on the fluid domain side and then the continuity of acoustic w and structure w displacement is ensured. Equation (1) leads to: a s 3

4 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century p x y L z t ρ V 0 w x y t x t y,,,,, p x y L z ρω w x y ikρcv 0 w x y ρv 0 w x y x y y,,,,, () where k the acoustic wavenumber, c the sound speed and w the structural displacement (the subscript is omitted). This coupling using the transverse momentum equation can also be written through the velocity v at the interface and leads to the equation (3). However, on the membrane equation, the displacement is not convected and v iωw. s s p x, y, L z, t ρ V 0 v x, y, t z t y p x y L z ρiωv x y ρv 0 v x y z y,,,, (3) Those formulations are equivalent only if there is no flow. Therefore, results shown below when no flow is considered, the coupling condition will not be specified. The acoustic and structure equations along the coupling and boundary conditions lead to the total operator presented in equation (4). p( x, y, z) p( x, y, z) Δ p( x, y, z) M ikm k p( x, y, z) 0 for ac.pressure y y p ik p i k k p inc at y = y 1 (plane wave from ) y p ik p 0 at y = y y 1 Δ w( x, y) kmw( x, y ) p x, y, Lz for themembrane T w( x, y1) w( x, y) w( x1, y) w( x, y) 0 dw d w p x, y, Lz ρω w( x, y) ikρcv0 ρv0 z dy dy (4) 4

5 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century This system uses a coupling through displacements but the same procedure applies for the other one. In equation (4), acoustic and structure equations are multiplied by test functions and integrated over the whole domain giving the weak form, itself integrated by part resulting in the variational form of the operator. The pressure field is obtained by computing the above operator in its variational form with a Finite Element home-made program (FEM). Results are presented in term of Insertion Loss (IL), defined by 0 10 with / without IL Log p p where the pressures with and without the membrane are taken at the end of the rigid, meshed guide (for the numerical computation), more precisely at the mid-point of it. The pressure "without membrane" is the incident pressure as long as we consider only its amplitude. 3 Analytical considerations The dispersion relation related to the configuration can be obtained by considering the structural and acoustic equation (in 1 dimension with the plane wave approximation) where the variation of mass flow rate is taken into account in the latter as a coupling condition. This system is represented in equation (5), the dimension L x represents the section / vibrating perimeter ratio. The structure is still in D and to reduce it to 1D, only the first transverse modal shape is considered for the membrane. Eventually, the structural equation is integrated along the vibrating perimeter. for the ac.pressure p( y) p( y) ρ 1 M ikm k p( y) iω V0 v( y) 0 y y Lx y 1 Δ w( x, y) kmw( x, y ) p( x, y ) for themembrane T (5) Using the structural approximation presented above, the dispersion relation (equation (6)) is then obtained by looking for solutions in ry e. m x 4 3 r 1 M r ikm r 1 M k k k ρv 0 ρω r iω ikm km kx k km kx 0 TLx TLx (6) 5

6 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century Figure : Celerity against frequency. Top: without flow, bottom left: M=0.3 with displacement coupling, bottom right: M=0.3 with velocity coupling The dispersion relation is of order four and four branches are expected from the dispersion graphs. Figure show them in term of celerity against frequency and, it is noticeable that without flow, four branches are present for a given frequency: two for each direction of propagation. From Figure, three main branches arise (in one direction): the first one is the "membrane" branch that tends to the wave celerity in the structure at high frequency (about 10 m/s here). The second one is a supersonic branch that tends to the sound celerity in the air at high frequency (anti- or convected celerity in case of flow). This branch appears beyond a specific frequency. With a flow, there is a "fifth" branch along the imaginary branch in both kind of coupling. As the dispersion relation is of order 4, it is surely an attenuated propagation. The coupling through displacement shows this "fifth" branch propagating in the main direction, although the coupling through velocity shows the branch propagating in the opposite direction. Also the flow doesn't modify the frequency where the supersonic branch appears. In the next section, results in term of TL obtained numerically will be presented. To be more confident on the code, the branches (real part only) are extracted from the pressure field along the y-axis (in the middle of the duct) using a Fast Fourier Transform (FFT) with the method presented in [7]. However, to allow a quicker computation (and avoid some memory-related problems) with the FFT as it needs a very long domain in the main propagation direction, the configuration has been modified for the sake of this comparison (only here) : Lx 4 cm instead of 8 cm. 6

7 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century Figure 3: Comparison between dispersion graphs obtained from analytical (dispersion relation) and numerical (FFT) way. Top : without flow, mid-left: M=0.3 (velocity coupl.) downstream, midright: M=0.3 upstream (velocity coupl.), bottom-left: M=0.3 downstream (displ. coupl.), bottomright: M=0.3 upstream (displ. coupl.) Figure 3 shows a acceptable agreement between analytical and numerical computations for the "membrane" branch. Although the same behavior is exhibited, there is a discrepancy for the supersonic branch but it should be explained by the difference in coupling procedure (mass flow or transverse pressure). Also, this discrepancy may be explained by the plane wave hypothesis (a strong hypothesis used in the analytical model) which is perhaps not verified in these region. In the two coupling approaches, the same remarks apply. However, the code fails to predict the "fifth" branch and, in case of an attenuated propagation, the wave would vanish too quickly to be measured by the FFT. 7

8 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century 4 Results 4.1 Without flow The attenuation computed numerically for the configuration of Figure 1 is firstly investigated numerically (FEM) without flow. Figure 4: Insertion Loss without flow for L y =30 cm (left) and L y =48 cm (right) In the Figure 4, the attenuation shown is very small (almost not significant). However, perhaps a suitable zone should be seen between 900 and 100 Hz which has no immediate relation with the branches behavior (Figure ). For the case when L =48 cm, the membrane itself as resonance peaks in this region at 897, 971, 1055 Hz and seem to correspond to the peaks shown in Figure 4 (if those value are not modified by the coupling). No clear attenuation are also visible in the zone where the imaginary branch exists. The model considered here is purely conservative and the attenuation would result from the evanescent (or attenuated propagation) wave and impedance breaking. Although, the lengthening of the membrane (Figure 4) doesn't exhibit a clear increase of attenuation which would have been the case for an evanescent mechanism. This is a further evidence that the branches are acting independently. To be more closely related to an experiment, damping should be taken into account [8]. It will also add attenuation as it does for a coupling with a plate [9]. 4. With flow For the same configuration (Figure 1), the IL is computed using equation (4) for both coupling condition: through displacement and through velocity. The comparison of branch using the FFT presented above allows to be more confident in the code, however further work are needed to verify and validate the results (especially with the displacement condition not shown here). For instance, in one hand, Figure 5 show the IL computed numerically and tendencies are not in good agreement with experiments found in literature. y 8

9 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century Figure 5: Numerical Insertion Loss for different flow velocities with coupling through velocity. On the other hand, Figure 6 and 7 give an analytical insight of what is expected: in both cases, the flow lowers the peaks. This effect have already been shown experimentally although it considers two facing membranes and cavities behind them [10]. The analytical method employed here uses the hypothesis presented above for the dispersion graphs. The peaks are lowered with more efficiency using the coupling through displacement model. An explanation could perhaps be found in the velocity linear and quadratic dependent term of the displacement coupling condition compared to the velocity linear-only dependant term in the equations (, 3). Figure 6: Analytical Insertion Loss for different flow velocities with coupling through velocity with a zoom ( Hz) on the right (top) or through the displacements (bottom). Black: M=0, Blue: M=0.15, Red: M=0.3. 9

10 nd International Congress on Acoustics, ICA 016 Buenos Aires 5 to 9 September, 016 Acoustics for the 1 st Century 5 Conclusion In the case of a convected guided acoustic wave coupled with a vibrating membrane, an investigation of the resulting attenuation is made by a numerical computation (using a sober finite element model). Two ways of taking into account the flow into the coupling are investigated: through the displacement and through the velocity. Also, to gain confidency in this simulation, a comparison of dispersion graphs (real branches only) is made between numerical and analytical approaches. However, the numerical computation is only partially validated and seem to need additional work to be coherent with the tendency acquired with the analytical solution, itself coherent with properties observed experimentally. For the present configuration where there are two distinct frequency regions, one with an attenuated and a propagating waves, the other with two propagating waves, the attenuation observed seems to arise from the impedance breaking brought by the section of the guide with vibrating walls. The addition of a flow reduce the attenuation performance of the configuration as shown experimentally from the literature [10]. This result is achieved by both coupling conditions, however the coupling through the displacement exhibits these behaviour more strongly. References [1] Morse P. M. and Ingard K. U. Theoritical acoustics, McGraw-Hill, New York, [] Martin V. Eléments d'acoustique générales, de quelques lieux communs de l'acoustique à une première maîtrise des champs sonores, PPUF, 007. [3] Astley R. J. and Cummings A. A finite element scheme for acoustic transmission through the walls of rectangular ducts : comparison with experiment, J. Sound Vib, Vol 9 (3), pp , [4] Liu Y., Choy Y. S., Huang L. and Cheng L. Reactive control of subsonic axial fan noise in a duct, J. Acoust. Soc. Am., Vol 136 (4), pp , 014. [5] Martin V., Cummings A. and Gronier C.. Discrimination of coupled structural/acoustic duct modes by active control: principles and experimental results, J. Sound Vib., Vol 74 (3-5), pp , 004. [6] Gauthier F., Gilbert J., Dalmont J.-P. and Pico Vila R. Wave propagation in a fluid filled rubber tube: theoritical and experimental results for korteweg's wave, Acta Acustica united with Acustica, Vol 93, pp , 007. [7] Meyer V., Martin V. Analytical Numerical comparison for the acoustic fields arising from a convected guided acoustic wave coupled with a yielding wall, Proceedings of the nd international congress on sound and vibration, Florence, Italy, 1-16 July 015. [8] Choy Y. S. and Huang L. Experimental studies of a drumlike silencer, J. Acoust. Soc. Am. Vol 11 (5), pp , 00. [9] Meyer V., Martin V., Zelmar P. Influence of the structural damping and/or the external acoustic load of a guided acoustic plane wave coupled with yielding walls, Proceedings of the 3rd international congress on sound and vibration, Athens, Greece, July 016. [10] Choy Y. S. and Huang L. Effect of flow on the drumlike silencer, J. Acoust. Soc. Am. Vol 118 (5), pp ,