Aeroacoustic simulation of the flow in a Helmholtz resonator

14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference) 5 - 7 May 2008, Vancouver, British Columbia Canada AIAA 2008-3005 Analysis of CFD and CA Solvers Requirements for Aeroacoustics Applications : case of the Helmholtz resonator Laurent Georges∗ and Philippe Geuzaine† Cenaero, rue des Frères Wright 29, 6041, Gosselies, Belgium Stéphane Caro‡ Free Field Technologies, rue Emile Francqui 1, 1435, Mont-Saint-Guibert, Belgium This paper reports on the improvement and validation of a computational framework developed for the simulation of aeroacoustics problems. An acoustic analogy is adopted and a in-house three-dimensional unstructured flow solver is coupled to the Actran/LA commercial finite element solver that uses a variational formulation of the Lighthill analogy. Numerical investigations are here performed in order to assess the appropriate level of accuracy required for both computational fluid dynamics and acoustics parts of this methodology to produce accurate noise predictions. As a matter of fact, these investigations are here focused on the noise radiated by a Helmholtz resonator placed in a duct. I. Introduction After a lot of work has been performed to reduce the noise generated by vibrating structures, aerodynamic noise sources are becoming more and more important in the transportation industry. Nowadays computational aeroacoustics (CAA) emerges as a viable solution to reduce the number of prototypes and tests, and thus the global design cost and duration. Aeroacoustics is governed by the compressible Navier-Stokes equations. Since a direct numerical simulation (DNS) of these equations is impractical for engineering problems, it thus calls for an alternative approach. The approach followed in this paper consists in using an acoustic analogy, as first proposed by Lighthill.1 Acoustic analogies rest on the assumption that noise generation and propagation are decoupled, that is, flow generated noise does not impact the internal dynamics of the flow. In practice, using an acoustic analogy is a two-step procedure. In the first step, an unsteady computational fluid dynamics (CFD) analysis is used to compute aerodynamic sources. The second step consists in computing the propagation and radiation of these aerodynamic sources (CA). I.A. CFD solver In order to perform the high Reynolds flow analysis for complex geometries, Cenaero has developed a parallel implicit solver for three-dimensional compressible flows on unstructured tetrahedral meshes. The method uses an edge-based hybrid finite volume and finite element discretization. It blends an upwind scheme for the convection fluxes based on Roe’s approximate Riemann solver and a piecewise linear reconstruction of the flow variables in each control volume, with a P1 finite element Galerkin approximation of the diffusive fluxes. This second-order accurate numerical scheme, which is representative of most numerical schemes used on unstructured meshes, was designed for Euler and Reynolds averaged Navier-Stokes (RANS) simulations, and therefore performs well for this type of flows. Since large eddy simulations (LES) have proved to perform better than RANS for complex phenomena like aerodynamic noise sources prediction, modifications were required by such a standard discretization on unstructured meshes to perform equally well for LES applications. ∗ Research Scientist, CFD-Multiphysics group Leader, CFD-Multiphysics group ‡ Research Scientist, AIAA Member † Group 1 of 10 American Institute of Aeronautics Copyright © 2008 by Copyright © 2008 by Cenaero. Published by the American Institute of Aeronauticsand and Astronautics Astronautics, Inc., with permission. The main issue with standard CFD schemes is the effect of the numerical dissipation introduced to stabilize the discretization of the convection term. It is well-known that this dissipation competes strongly with the effect of the subgrid scale (SGS) model. An easy way to circumvent this problem is to resort to central schemes. Unfortunately, central schemes are in general unstable as the Reynolds number is increased. This behavior is due to the spurious discrete kinetic energy injection of those schemes. At low Reynolds numbers, this injection can be counterbalanced by the diffusion term. As a consequence, the simulation remains stable but results can be unphysical. Nevertheless, the simulation becomes rapidly unstable at high Reynolds numbers. A possible way to reach stability is to ensure that the central scheme conserves the discrete kinetic energy. In this paper, the central scheme developed in Ref. 2 for compressible shock-free flows is adopted and further validated for aeroacoustic applications. I.B. Noise propagation solver Within an acoustic analogy framework, a popular approach to the propagation and radiation of the aerodynamic sources is to rely on explicit integral methods, amongst which the most famous is the Ffowcs-Williams and Hawkings equation. There are however several limitations to such techniques since they are practically limited to pure exterior radiation problems as they can hardly be used in interior problems (e.g. in ducts). Furthermore, they require compressible CFD simulations, although the vortex structures could mostly be computed with incompressible CFD well enough. These limitations have pushed Free Field Technologies to implement an alternate method3 in the finite element code Actran/LA. The method is based on the variational formulation of the Lighthill equation, is designed to be used for exterior or interior problems with or without liners, and has been shown to possess the potential to handle industrial problems.4–6 I.C. Objectives The objective of the present contribution is to investigate the accuracy of the proposed CAA methodology. The main benchmark used for these investigations is the noise generated by a Helmholtz resonator placed in a duct, reported on Sec. III. Previous tests7, 8 have already proved the ability of the methodology to properly capture the peak of resonance (narrow-band noise), in terms of frequency and amplitude. Nevertheless, the broad-band far-field noise was rather under-predicted. These investigations are here followed in order to further analyse the computed broad-band noise and to determin the relevant parameters of the methodology that drive the solution quality. II. Computational aeroacoustics methodology The theory3, 9, 10 behind the formulation used in Actran/LA is briefly summarized hereafter. Starting from the mass and momentum conservation equations, it is possible to derive Lighthill’s equation without any assumption, as in the beginning of the original paper.1 Then, some classical assumptions, valid only in the case of a low Mach number and a high Reynolds number flow where isentropic assumptions are reasonable from an acoustic point of view, lead to dramatic simplifications. The final equation is a true wave equation whose right-hand side term is the simplified Lighthill’s tensor ∂ 2 Tij ∂ 2 ρa ∂ 2 ρa = , − a20 2 ∂t ∂xi ∂xi ∂xi ∂xj (1) Tij ≃ ρ0 vi vj . (2) with In the frequency domain and adapted to a finite element framework, the formulation becomes3, 11 ! Z ∂ ρ̃ ∂δρ ∂δρ 1 ∂ T̃ a a a ij −k 2 ρ̃a δρa + + 2 dΩ = 0 ∀ δρa . ∂xi ∂xi a0 ∂xj ∂xi Ω (3) The only missing quantity is the source term, represented by the divergence of the Lighthill tensor in the frequency domain. This quantity is computed in the time domain by the CFD solver and a Fourier transform is then used before the field is passed to Actran/LA. The Fourier transform is performed using a dedicated package which comes with the standard distribution of Actran/LA. The whole computational aeroacoustics process can be summarized as follows: 2 of 10 American Institute of Aeronautics and Astronautics 1. The user produces a file with the node coordinates of the subset of the acoustic mesh where the sources must to be accounted for; 2. The CFD solver reads this file; then, during the unsteady CFD simulation, at each time step, it computes and writes the divergence of the Lighthill tensor at all these nodes locations; 3. The user does the Fourier transform of the result, using filters if needed; 4. The user launches the Actran/LA simulation, which gives a direct access to all acoustic fields in the finite and infinite elements, including some energy indications. The files produced for Actran/LA by the CFD solver are written in the Hierarchical Data Format (HDF), an open-source format used to store named matrices and developed by NCSA. It is an OS-independent binary format. The Scientific DataSets of HDF version 4 is used. III. Flow in a Helmholtz resonator To illustrate and validate the computational aeroacoustic methodology presented in this paper, the simulation of the noise radiated by a Helmholtz resonator placed in a duct is considered (see Figs. 1). This testcase has been studied experimentally in the reverberation chamber of the acoustic laboratory at Behr.5 The experimental results show that the resonance frequency, fexp , which is independent of the inlet flow velocity, is equal to 358 Hz which is consistent with the analytical formula, fth given in Ref. 5 s c0 πr2 fth = (4) 2π V (L + ∆L) where c0 is the speed of sound, r, the tube internal radius of 6 mm, L, the tube length of 50 mm, ∆L, an orifice correction and V is the volume of the resonator (a cube with 35 mm edges and 1.5 mm thickness walls). Ø 15 140 38 500 140 (a) Global view (b) close-up Figure 1. Geometry of the Helmholtz resonator placed in a duct III.A. CFD computations The baseline tetrahedral mesh used for the CFD computations, here called CFD1, is shown in Figs. 2. It contains 1.087 M nodes and is refined according to the physics of the Helmholtz resonator, see Tab. 2. In fact, a shear layer is generated by the upstream part of tube that is inclined towards the domain inlet. This is pictured by Fig. 2(d) representing the instantaneous vorticity norm in the symmetry plane of the resonator. This shear layer is unstable and acts as the perturbation, as the input, for the resonance of the cavity. The pressure inside the tube is then fluctuating in accordance with its natural frequency. Furthermore, if this frequency is close the instability frequency of the shear-layer, the pressure pulsation of the tube can trigger 3 of 10 American Institute of Aeronautics and Astronautics Table 1. Parameters of the air flow around a Helmholtz resonator placed in a duct Flow parameter Inlet velocity Inlet temperature Outlet pressure Dynamic viscosity Thermal conductivity Theoretical resonance frequency Measured resonance frequency5 Symbol U0 T0 p0 µ κ fth fexp Value 7.787 [m/s] 293.15 [K] 101300 [Pa] 1.8 10−5 [kg/ms] 2.51 10−2 [W/mK] 360 [Hz] 358 [Hz] the shear-layer instability and amplify it. Finally, this leads to an amplification of the power at the resonance frequency. It is therefore important to properly capture the shear-layer instability near the resonator neck. This is done using a well refined mesh in this region, as depicted in Fig. 2(b). Beside the phenomena near the resonator neck, the global flow around the resonator is representative of a bluff body flow (here, the flow around a wall-mounted cylinder). The mesh is refined in accordance with the physics of such flow. In fact, the Reynolds number based on the resonator diameter, ReD , is equal to 7818. In this regime, the laminar boundary layers around the tube detach and form shear-layers that are unstable and finally lead to a fully turbulent wake. An anisotropic boundary layer mesh is thus generated from the tube external surface to capture the boundary layers properly, see e.g. Fig. 2(b). The mesh is also refined in the wake and coarsens with the distance from the resonator, see Figs. 2(a) and 2(c). A finer mesh of 2.467 M nodes has also been considered. This one, termed CDF2, differs from the baseline mesh CFD1 by halving the characteristic size of the elements in the medium refinement zone, see Tab. 2. Table 2. Parameters of the meshes used for the CFD and acoustic propagation part : the definition of the refinement zone as well as their denomination is defined in Fig. 2 Global size [mm] Large refinement (L) [mm] Medium refinement (M) [mm] Small refinement (S) [mm] X-Small refinement (XS) [mm] XX-Small refinement (XXS) [mm] Boundary layer mesh Mesh acoustic cutt-off frequency [Hz] Number of nodes CFD1 (baseline) 5.0 2.5 1.25 0.625 0.3 0.1 Yes No 1087 k CFD2 (fine) 5.0 2.5 0.625 0.625 0.3 0.1 Yes No 2467 k CA1 (coarse) 12.8 6.4 3.2 3.2 0.8 No No ∼ 4.5 k 59 k CA2 (medium) 12.8 6.4 1.6 1.6 0.4 No No ∼ 4.5 k 268 k CA3 (fine) 12.8 6.4 1.25 0.625 0.3 No No ∼ 4.5 k 706 k CA4 (uniform) 9.6 No 2.4 No No No No ∼ 6.0 k 75 k Given the medium Reynolds number under investigation, the simulation can be performed using wallresolved Large Eddy Simulation (LES). The selected subgrid scale (SGS) model is the WALE model of Nicoud and Ducros.12 The spatial discretization of the convective term is based on the kinetic-energy conserving scheme supplemented by an hyperdiffusion term, while the time-integration uses the mass matrix and is performed by the second-order accurate three-point backward difference scheme with a time-step equal to 2 × 10−5 s. This leads to ∼ 140 times steps by tube resonance period. The simulation was performed during 10000 time steps in the statistical steady state, giving a total simulation time of 0.2 s. The Mach number, M0 , is equal to 0.0227. Even though this Mach number is rather low, the non-linear system to be solved at each time step was not particularly stiff so that no special strategy was followed to face this low-Mach application. No-slip conditions were applied at the external surface of the tube as well as at the lateral surface of the tube inlet. From Fig. 2(d), it’s clearly visible that a significant part of vortical structures 4 of 10 American Institute of Aeronautics and Astronautics (a) Side view (along the symmetry plane) (b) Close-up on the resonator neck (c) Top View (cut plane close to the bottom wall) (d) Vorticity near the resonator neck Figure 2. Definition of the mesh refinement zones on the baseline CFD mesh : dashed red box termed large refinement(L), dashed-dotted blue box termed medium refinement(M), dashed-dotted-dotted magenta box termed small refinement (S), dashed-dotted-dotted-dotted green box term X-small refinement(XS) and solid red box termed XX-small refinement(XXS) are entering the tube. The boundary conditions applied for the interior walls of the resonator tube are thus important. As the mesh inside the resonator tube is not fine enough to properly capture the viscous interaction of the vortices with the inside wall, a Reichardt’s wall function13 was there applied. Finally, slip boundary conditions were imposed for the duct walls as the viscous effects near these walls are not supposed to play a role in the noise generation. Furthermore, capturing these viscous effects would impose to refine the mesh near the duct walls and would eventually lead to a drastic increase of the number of mesh points. 100 Pressure [Pa] 50 0 -50 -100 0.000 0.005 0.010 0.015 0.020 time [s] 0.025 0.030 0.035 0.040 Figure 3. Time evolution of the pressure variation in center of the resonator : results obtained using the baseline mesh CFD1 (solid line) and using the fine mesh CFD2 (dashed line) A complete resonance cycle is illustrated using successive snapshots of the vorticity norm field along with the corresponding pressure field, see Figs. 4. The pressure is monitored in the center of the tube during the 5 of 10 American Institute of Aeronautics and Astronautics simulation, see Fig. 3. A Fourier transform of this signal reveals that the computed resonance frequency value is equal to 350 Hz, which is in excellent agreement with the experimental and analytical values, see Tab. 1. Furthermore, the refinement from mesh CFD1 to CFD2 does alter this frequency but has an influence on the amplitude of the pressure fluctuations. All the tests performed during the presend study showed that the computed resonance frequency is always close to 350 Hz and never converges towards the theoretical value, 360 Hz. An hypothesis can be proposed for this behaviour. As explained previously, a significant amount of vortical structures are entering the resonator tube, see e.g. Figs. 4. The presence of such structures obstructing the tube are not integrated in the theoretical model, and may partially reduce the surface that is effective for the resonance phenomenon. In fact, numerical tests have shown, that the resonance frequency is highly sensitive to the boundary condition applied at the tube internal surfaces : applying pure no-slip conditions, albeit the coarse near-wall mesh resolution, gave an under-estimation of 434 Hz. This confirms that vortical structures, generated by the impingement of the oscillating shear-layer and partially entering the tube, have a major impact on the computed frequency. (a) Maximum of pressure (b) Relaxation phase (c) Minimum of pressure (d) Compression phase Figure 4. Illustration of a complete resonance cycle : snapshots of the instantaneous pressure field as well as the related vorticity norm field are reported during four differents stages of the cycle. III.B. Acoustic computations Four differents meshes were used for the acoustic propagation simulations. The first three meshes were basically defined using the the same refinement zones as for the CFD meshes, see Tab. 2 and Figs. 5. This ensures that the ratio between the CFD and CA mesh element sizes is constant within a given refinement zone. In this way, the quality of the interpolation of the sources from the a CFD mesh to an acoustic mesh is almost constant throughout the source region. This source region, defined as the region where the Lighthill source term is evaluated, is limited to the medium refinement zone (M), see Fig. 2. Let’s mention that no boundary layer mesh is present for the acoustic meshes so that the ratio between meshes can vary in this region. The first mesh, termed CA1, is a coarse mesh of 59 k nodes. The second mesh, termed CA2, is more refined in the source region and is composed of 268 k nodes. The third mesh, termed CA3, is taken equal to the baseline CFD1 mesh in the source region and has 706 k nodes. It enables to analyse the results without introducing error due to the interpolation. Futhermore, this phase can be analysed using Fig. 6 where the instantaneous source term norm is pictured : the initial field on the baseline CFD1 mesh is interpolated to three different acoustics meshes. One clearly notice that mesh CA1 is so coarse that an important part of 6 of 10 American Institute of Aeronautics and Astronautics the coherence of the sources is lost, see Fig. 6(b). On the contrary, this coherence is properly retrieved by the medium mesh , Fig. 6(c), while the fine mesh reproduces, by definition, the source field exactly, Fig. 6(d). The last mesh, termed CA4 and composed of 75 k nodes, is taken uniform throughout the source region. The difference between CFD1 and CA4 mesh sizes vary drastically within the source region and is more prononced near the resonator neck. Figure 5. Computational domain used for the acoustic propagation : the domain is pictured by the triangular surface mesh. (a) CFD1 (b) CA1 (c) CA2 (d) CA3 Figure 6. Analysis of the interpolation procedure from the baseline CFD mesh to the three different acoustic meshes : visualization using the instantaneous divergence norm of the Lighthill tensor. A modal basis is applied at the duct inlet while infinite elements applied at the oulet boundary enables to model a free field radiation problem. The monitored value that is compared to the experiment5 is the power radiated through the outlet boundary. Acoustic sources were sampled every 10 time steps, corresponding to a maximum frequency of 2500 Hz, the total sampling time is 0.2 s and gives a minimum frequency of 5 Hz. Results are reported on Figs. 7. The first graph, Fig. 7(a), compares the experimental data with the radiated power computed using the baseline CFD1 mesh and the first three different acoustic meshes. As expected, the peak of resonance is 7 of 10 American Institute of Aeronautics and Astronautics well reproduced with a value of 68 dB at 258 Hz for the experiment and 62 dB at 250 Hz for the three CAA simulations. The coarse mesh CA1 gives a broad-band spectrum that is in pretty good agreement with the experiment until 1 kHz. Nevertheless, the mesh convergence study shows that the computed broad-band noise collapses for high-frequencies along with the mesh refinement. Furthermore, similar results for the medium (CA2) and fine (CA3) meshes proves that the acoustic problem is grid-converged. Nevertheless, the computed powers always remain close to the experiment for low-frequencies located before the peak of resonance. Results using CFD1 and the uniform mesh, CA4, are reported on Fig. 7(b). As the CA4 mesh does not follow the CFD1 mesh refinement, the difference in mesh size is important near the resonator neck. This leads to large interpolation errors and the lost of the sources coherence. In this case, eventhough the mesh CA4 is finer that the coarse mesh CA1, the amplitude of the resonance peak is overestimated drastically (78 dB for the 75 nodes CA4 mesh, instead of 62 dB for the 59 k nodes CA1 mesh). The third graph, Fig. 7(c), compares the experimental data with the radiated power computed using different CFD and the same medium acoustic mesh (CA2). The peak of resonance is still located at 250 Hz but the amplitude is better captured using the fine CFD mesh, with a value of 65 dB : this tendancy is in good accordance with the amplitude of the fluctuating pressure that indeed increased for the fine CFD2, see Fig. 3. The broad-band noise is identical between both simulations, but for the fine CFD2, the low-frequency broad-band noise is reduced and departs from the experimental curve. It is worth mentionning that the mesh refinement in the wake, that differs the fine mesh (CFD2) from the baseline one (CFD1), did not increase the global level of the broad-band noise (on the contrary, the radiated power is reduced for low-frequencies). In fact, one would expect that the broad-band noise originates from the wake of the resonator tube and is almost independant of the narrow-band resonance phenomenon. Said otherwise, the broad-band noise should be roughly equivalent to the noise generated by a wall-mounted cylinder. The mesh refinement in the wake, performed for the fine mesh CFD2, should have captured the noise sources more properly so that one would finally expect to find an increased broad-band noise for the finer CFD2 mesh. This is not the case here as the global broad-band level is almost unchanged. A possible reason is that the CFD2 is again not fine enough but increasing the mesh size would be very expensive in terms of computational resources. Furthermore, the quality of the experiment is also questionable and it is finally difficult to say whether one should expect better results for this benchmark. An argument in favor of this last statement is that all computed powers are more or less similar, except for the very coarse meshes, CA1 and CA4, where the noise sources coherence is clearly lost (see Fig. 6(b)). Finally, an important conclusion can be derived from the present study. The solution is highly sensitive to the quality of the noise sources representation on the acoustic mesh. A good representation can be achieved if the ratio between the local CA mesh size and the local CFD mesh size is maintened to a reasonable low value. This can be done using an acoustic mesh refinement that is in accordance with the CFD mesh refinement (as it was realised for CA2 and CA3 meshes). Said otherwise, the acoustic mesh should not only be designed according to its cutt-off frequency, that ensures the correct noise propagation, but also to preserve the coherence of the noise sources. In the present study, a simple uniform acoustic mesh, CA4, failled to reach this objective and this leads to a significant overshoot of the resonance peak. IV. Conclusion The present contribution aimed to further validate the hybrid CAA approach based on the LightHill analogy and coupling the Cenaero CFD code,Cenaero Argo, to the commercial acoustic solver Actran/LAfrom Free Field Technologies (FFT). The selected benchmark is the noise generated by a Helmholtz resonator placed in a duct. As already confirmed in previous investigations, the narrow-band noise corresponding the peak of resonance is well predicted and is pretty good agreement with the experiments at Behr.5 Furthermore, the present study improved the resolution of this peak in terms of amplitude. Nevertheless, the broad-brand noise predicted from the different computations still differs significantly from the experiment. None of the proposed mesh refinements (for the CFD as well as for CA meshes) managed to increase the mean power level of the broad-band noise. Of course, further investigations could be devoted to the present benchmark. Nonetheless, the overall accuracy of the experimental procedure is also questionable. A better approach could be to investigate the present methology on a full-validated benchmark. It raises the question of the existence of such absolute reference that could validate thouroughly a CAA hybrid methodology. Finally, an important conclusion can be derived from the present study. The solution is highly sensitive to 8 of 10 American Institute of Aeronautics and Astronautics (a) CFD1 and CA meshes (b) CFD1 and uniform CA mesh (c) all CFD with CA2 mesh Figure 7. Radiated power of the flow in a Helmholtz resonator : on top, the power obtained using the baseline CFD1 mesh and the four different acoustic meshes (on the left, black curve for the coarse CA1 mesh, red curve for the medium CA2 mesh and the blue curve for the fine CA3 mesh, and on the right, blue curve for CA4 uniform mesh), on the bottom, the power obtained using the medium acoustic CA2 mesh and the two diffent CFD (red curve for the baseline CFD1 mesh and blue curve for the fine CFD2); finally, experimental data are reported in magenta. the quality of the noise sources representation on the acoustic mesh. A good representation can be achieved if the ratio between the local CA mesh size and the local CFD mesh size is maintened to a reasonable low value. As a conclusion, the acoustic mesh should not only be designed according to its cutt-off frequency, that ensures the correct noise propagation, but also to preserve the coherence of the noise sources. Acknowledgments The authors want to acknowlegde Pierre Dulieu that produced a part of the present work during its Master’s thesis at the Université catholique de Louvain (UCL). This contribution, performed at Cenaero, was achieved in 2007 and supervised by the professor Grégoire Winckelmans (UCL). Besides, the first two authors acknowledge the support by the Walloon Region and the European funds ERDF and ESF under contract N◦ EP1A122030000102. References 1 Lighthill, 2 Georges, M., “On Sound Generated Aerodynamically,” Proc. Roy. Soc. (London), Vol. A 211, 1952. L., Winckelmans, G. S., and Geuzaine, P., “Improving Shock-free Compressible RANS Solvers for LES on 9 of 10 American Institute of Aeronautics and Astronautics Unstructured Meshes,” Journal of Computational and Applied Mathematics, Vol. 215, No. 2, 2008, pp. 419–428. 3 Caro, S., Ploumhans, P., and Gallez, X., “Implementation of Lighthill’s Acoustic Analogy in a Finite/Infinite Elements Framework,” AIAA paper 2004-2891. 4 Mendonca, F., Read, A., Caro, S., Debatin, K., and Caruelle, B., “Aeroacoustic Simulation of Double Diaphragm Orifices in an Aircraft Climate Control System,” AIAA paper 2005-2976. 5 Caro, S., Ploumhans, P., Brotz, F., Schrumpf, M., Mendonca, F., and Read, A., “Aeroacoustic Simulation of the Noise radiated by an Helmholtz Resonator placed in a Duct,” AIAA paper 2005-3067. 6 Caro, S., Ploumhans, P., Morgenthaler, V., and Mathey, V., “Identification of the Appropriate Parameters for Accurate CAA,” AIAA paper 2005-2991. 7 Georges, L., Winckelmans, G. S., Caro, S., and Geuzaine, P., “Aeroacoustic simulation of the flow in a Helmholtz resonator,” 4th International Conference on Computational Fluid Dynamics, Ghent, Belgium, July 2006. 8 Geuzaine, P., Bogaerts, S., Georges, L., Hillewaert, K., and Caro, S., “Improving an Unstructured CFD Solver for Aeroacoustics Applications,” AIAA paper 2006-2473. 9 Oberai, A., Ronaldkin, F., and Hughes, T., “Computational Procedures for Determining Structural-Acoustic Response due to Hydrodynamic Sources,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, 2000, pp. 345–361. 10 Oberai, A., Ronaldkin, F., and Hughes, T., “Computation of Trailing-Edge Noise due to Turbulent Flow over an Airfoil,” AIAA Journal, Vol. 40, 2002, pp. 2206–2216. 11 Free-Field-Technologies-S.A., Actran 2005 Aeroacoustic Solutions: Actran/TM and Actran/LA - User’s Manual, 16 place de l’Université, 1348 Louvain-la-Neuve, Belgium, 2005. 12 Nicoud, F. and Ducros, F., “Subgrid-scale stress modelling based on the square of the velocity gradient tensor,” Flow, Turbulence and Combustion, Vol. 62, 1999, pp. 183–200. 13 Reichardt, H., “Vollständige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen,” Zeitschrift für Angewandte Mathematik und Mechanik , Vol. 31, 1951, pp. 208–219. 10 of 10 American Institute of Aeronautics and Astronautics