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spanwise near-field expression (9), neglecting chorwise effects and taking spanwise effects only in the scaling term defined in Eq. (7) is compared to the direct numerical integration of formulation (5), without any geometrical assumptions. The maximum difference between formulations is up to 5 dB for z < d/20, where chordwise effects start to appear, but improves highly the solution compared to the far-field Amiet’s solution. If a direct numerical integration of formulation (5) is performed with similar assumptions than in formulation (9), identical results are obtained between both formulations. The spanwise effects neglected in the exponential term to obtain formulation (9) have then a negligeable effect on the sound results. The near-field expression (9) is attractive for geometrical near-field computation due to its facility of implementation and solution robustness, compared to direct integration requiring higher computational time to obtain similar accuracy due to the low convergence of Monte Carlo techniques. Nevertheless the direct integration of the Amiet’s general formulation (5) is a valuable tool to test the different assumptions/simplifications that could be introduced in the Amiet’s theory and to quantify the level of accuracy relative to the different assumptions. 3.2 Influence of acoustical far-field assumptions The same test case is used to evaluate the acoustical farfield approximation introduced in formulation (5) compared to the general formulation (4). Figure 3 (bottom) shows the results obtained using Amiet’s theory at different observer distances from the airfoil including geometrical near-field terms and acoustical near-field terms, together with combinations of the different simplifications. A first observation is that the differences are appearing, as expected, for a distance comparable to the magnitude of the acoustical wavelength. The difference observed using acoustical near-field terms on formulation (11) is around 10 dB close to the airfoil while the difference observed on formulation (4) is around 6-7 dB. It seems then that the influence of acoustical near-field terms are attenuated when geometrical near-field terms are taken into account. Finally, we should notice that, for the particular frequency selected (f = 2000 Hz), avoiding the use of acoustical near-field terms in the computation can be acceptable while for smaller frequencies, the deviations could increase dramatically, the influence of the acoustical near-field terms increasing with lower frequencies. 4 Concluding remarks In this work, we have investigated the influence of the geometrical and acoustical near-field effects that could be important in case of observers close to the airfoil surface. Those effects have been studied through an implementation of a generalised formulation of Amiet’s theory using the Monte Carlo technique to solve the five dimensional integral involved in the formulation. This generalised formulation revealed to be an efficient tool to verify the assumptions and estimate the potential of simplified formulations based on Amiet’s theory taking geometrical near-field effects into account. Nevertheless, the Monte Carlo technique used in the generalised formulation requires a large computational effort to obtain accurate results, but the implementation would be easily used in parallel to improve its efficiency. Acknowledgements The work presented in this paper has been sponsored by the European Commission under the framework of the FP7 Collaborative Project ECOQUEST (Grant Agreement no 233541). The authors are thankful to Prof. Michel Roger for fruitful discussions on the near-field generalization of Amiet’s theory. References [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Courier Dover Publications, 1964. [2] J. J. Adamczyk. The Passage of an Infinite Swept Airfoil Through an Oblique Gust, 1974. CR 2395. [3] R. K. Amiet. Acoustic Radiation from an Airfoil in a Turbulent Stream. Journal of Sound and Vibration, 41(4):407–420, 1975. [4] R. K. Amiet. Noise due to Turbulent Flow past a Trailing Edge. Journal of Sound and Vibration, 47(3):387–393, 1976. [5] T. F. Brooks, S. B. Pope, and M. A. Marcolini. Airfoil Self Noise and Prediction. NASA Reference Publication, July 1989. [6] S. Caro and S. Moreau. Aeroacoustic Modeling of Low Pressure Axial Flow Fans. In 6th AIAA/CEAS Aeroacoustics Conference, 2000. AIAA-2000-22094. [7] C. Carton de Wiart, P. Geuzaine, Y. Detandt, J. Manera, S. Caro, Y. Marichal, and G. Winckelmans. Validation of a Hybrid CAA Method: Noise Generated by a Flap in a Simplified HVAC Duct. In 16th AIAA/CEAS Aeroacoustics Conference, 2010. AIAA-2010-3995. [8] J. Christophe. Application of Hybrid Methods to High Frequency Aeroacoustics. von Karman Institute for Fluid Dynamics, 2011. [9] N. Curle. The Influence of Solid Boundaries upon Aerodynamic Sound. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 231(1187):505–514, 1955. [10] S. A. L. Glegg, S. M. Baxter, and A. G. Glendinning. The Prediction of Broadband Noise from Wind Turbines. Journal of Sound and Vibration, 118(2):217– 239, 1987. [11] M. E. Goldstein. Aeroacoustics. McGraw-Hill Inc., 1976. [12] J. O. Hinze. Turbulence. McGraw-Hill Inc., 1975. [13] K. Kucukcoskun, J. Christophe, J. Anthoine, C. Schram, and M. Tournour. An Extension of Amietï£¡s Theory for Spanwise-Varying Incident Turbulence and Broadband Noise Scattering Using a Boundary Element Method. In 16th AIAA/CEAS Aeroacoustics Conference, 2010. AIAA-2010-3987. [14] K. Kucukcoskun, J. Christophe, C. Schram, and M. Tournour. Broadband Scattering of the Incoming Turbulence Noise of a Stationary Airfoil: Experimental Validation of the Semi-Analytical Model. Submitted to AIAA J. 48 ERCOFTAC Bulletin 90
[15] M. A. Mac Donald, Y. Nishi, and J. Gullbrand. Notebook Blower Inlet Flows : Experiments and Modeling. In Fan Noise 2007, 2007. [16] O. M. Mohamud and P. Johnson. Broadband Noise Source Models as Aeroacoustic Tools in Designing Low NVH HVAC Ducts. In SAE Technical Paper Series, 2006. 2006-01-1192. [17] S. Moreau and M. Roger. Back-Scattering Correction and Further Extensions of Amietï£¡s Trailing- Edge Noise Model.Part II : Application. Journal of Sound and Vibration, 323:397–425, 2009. [18] S. Moreau, M. Roger, and V. Jurdic. Effect of Angle of Attack and Airfoil Shape on Turbulence- Interaction Noise. In 11th AIAA/CEAS Aeroacoustics Conference, 2005. AIAA-2005-2973. [19] R. W. Paterson and R. K. Amiet. Acoustic Radiation and Surface Pressure Characteristics of an Airfoil Due to Incident Turbulence, 1976. CR 2733. [20] S. Pérennès and M. Roger. Aerodynamic Noise of a Two-Dimensional Wing with High-Lift Devices. In 4th AIAA/CEAS Aeroacoustics Conference, 1998. AIAA-1998-2338. [21] M. Roger and S. Moreau. Back-Scattering Correction and Further Extensions of Amietï£¡s Trailing- Edge Noise Model. Part 1: Theory. Journal of Sound and Vibration, 286:477–506, 2005. [22] B. A. Singer, D. P. Lockard, and K. S. Brentner. Computational Aeroacoustic analysis of Slat Trailing-Edge Flow. AIAA Journal, 38(9):1558– 1564, September 2000. [23] M. Terracol, E. Labourasse, E. Manoha, and P. Sagaut. Simulation of the 3D Unsteady Flow in a Slat Cove for Noise Prediction. In 9th AIAA/CEAS Aeroacoustics Conference, 2003. AIAA-2003.3110. ERCOFTAC Bulletin 90 49
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Page 5 and 6: Special Issue Coordinated by SIG-39
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