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An analytical correction to Amiet’s solution of airfoil leading-edge noise in non-uniform mean flows

Published online by Cambridge University Press:  12 November 2019

Siyang Zhong Open the ORCID record for Siyang Zhong [Opens in a new window] ,
Xin Zhang ,
Bo Peng  and
Xun Huang
Siyang Zhong*
Affiliation:
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Hong Kong, PR China Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, PR China
Xin Zhang*
Affiliation:
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Hong Kong, PR China HKUST–Shenzhen Research Institute, Shenzhen, PR China
Bo Peng
Affiliation:
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Hong Kong, PR China
Xun Huang
Affiliation:
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Hong Kong, PR China State Key Laboratory of Turbulence and Complex Systems, Department of Aeronautics and Astronautics, College of Engineering, Peking University, Beijing, PR China
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Gust/turbulence–leading edge interaction is a significant source of airfoil broadband noise. An approach often used to predict the sound is based on Amiet’s flat-plate solution. Analytical studies have been conducted to investigate the influences of airfoil geometries, non-uniform mean flows and turbulence statistics, which, however, were often too convoluted. In this work, the problem is revisited by proposing simple corrections to the standard flat-plate solution to account for the effect of non-uniform mean flows of real airfoils. A key step in the method is to use a new space–time transformation that is analogous to the Prandtl–Glauert transformation to simplify the sound governing equation with spatially varying coefficients to a classical wave equation, which is then solved using the Schwarzschild technique as in Amiet’s solution. The impacts of Mach number, wavenumber and airfoil geometry on the prediction accuracy are investigated for both single-frequency and broadband cases, and the results are compared against high-fidelity simulations. It predicts the sound reduction by the airfoil thickness, and reveals that the reduction is caused by the non-uniform streamwise velocity. The limitations of the model are discussed and the approximation errors are estimated. In general, the prediction error increases with the airfoil thickness, the sound frequency and the flow Mach number. Nevertheless, in all cases studied in this work, the proposed correction can effectively improve the prediction accuracy of the flat-plate solution much more efficiently compared to numerical solutions of the Euler equations using computational aeroacoustics.

JFM classification

Acoustics: Aeroacoustics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 882 , 10 January 2020 , A29
Copyright
© 2019 Cambridge University Press 

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References

Adamczyk, J. J. 1974 Passage of a swept airfoil through an oblique gust. J. Aircraft 11 (5), 281–287.Google Scholar
Amiet, R. K. 1974 Compressibility effects in unsteady thin-airfoil theory. AIAA J. 12 (2), 252–255.Google Scholar
Amiet, R. K. 1975 Acoustic radiation from an airfoil in a turbulent stream. J. Sound Vib. 41 (4), 407–420.Google Scholar
Amiet, R. K. 1976a High frequency thin-airfoil theory for subsonic flow. AIAA J. 14 (8), 1076–1082.Google Scholar
Amiet, R. K. 1976b Low-frequency approximations in unsteady small perturbation subsonic flows. J. Fluid Mech. 75 (3), 545–552.Google Scholar
Amiet, R. K.1989 Noise produced by turbulent flow into a rotor: theory manual for noise calculation. NASA Tech. Rep. 181788.Google Scholar
Amiet, R. K. & Sears, W. R. 1970 The aerodynamic noise of small-perturbation subsonic flows. J. Fluid Mech. 44 (2), 227–235.Google Scholar
Ashcroft, G. & Zhang, X. 2003 Optimized prefactored compact schemes. J. Comput. Phys. 190 (2), 459–477.Google Scholar
Atassi, H. M. 1984 The sears problem for a lifting airfoil revisited-new results. J. Fluid Mech. 141, 109–122.Google Scholar
Atassi, H. M., Dusey, M. & Davis, C. M. 1993a Acoustic radiation from a thin airfoil in non-uniform subsonic flows. AIAA J. 31 (1), 12–19.Google Scholar
Atassi, H. M., Fang, J. & Patrick, S. 1993b Direct calculation of sound radiated from bodies in nonuniform flows. Trans. ASME J. Fluids Engng 115 (4), 573–579.Google Scholar
Atassi, H. M. & Grzedzinski, J. 1989 Unsteady disturbances of streaming motions around bodies. J. Fluid Mech. 209, 385–403.Google Scholar
Atassi, H. M., Subramaniam, S. & Scott, J.1990 Acoustic radiation from lifting airfoils in compressible subsonic flow. AIAA Paper 1990-3991.Google Scholar
Ayton, L. J. 2016 An analytic solution for gust–aerofoil interaction noise including effects of geometry. IMA J. Appl. Maths 82 (2), 280–304.Google Scholar
Ayton, L. J. & Chaitanya, P. 2017 Analytical and experimental investigation into the effects of leading-edge radius on gust–aerofoil interaction noise. J. Fluid Mech. 829, 780–808.Google Scholar
Ayton, L. J. & Peake, N. 2015 On high-frequency sound generated by gust–aerofoil interaction in shear flow. J. Fluid Mech. 766, 297–325.Google Scholar
Ayton, L. J. & Peake, N. 2016 Interaction of turbulence with the leading-edge stagnation point of a thin aerofoil. J. Fluid Mech. 798, 436–456.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7 (1), 83–103.Google Scholar
Blandeau, V. P., Joseph, P. F., Jenkins, G. & Powles, C. J. 2011 Comparison of sound power radiation from isolated airfoils and cascades in a turbulent flow. J. Acoust. Soc. Am. 129 (6), 3521–3530.Google Scholar
Christophe, J.2011 Application of hybrid methods to high frequency aeroacoustics. PhD thesis, Université libre de Bruxelles.Google Scholar
Christophe, J., Anthoine, J. & Moreau, S. 2009 Amiet’s theory in spanwise-varying flow conditions. AIAA J. 47 (3), 788–790.Google Scholar
Crighton, D. G. 1975 Basic principles of aerodynamic noise generation. Prog. Aeronaut. Sci. 16 (1), 31–96.Google Scholar
Curle, N. 1955 The influence of solid boundaries upon aerodynamic sound. Proc. R. Soc. Lond. A 231 (1187), 505–514.Google Scholar
Devenport, W. J., Staubs, J. K. & Glegg, S. A. L. 2010 Sound radiation from real airfoils in turbulence. J. Sound Vib. 329 (17), 3470–3483.Google Scholar
Drela, M. 1989 XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils. Springer.Google Scholar
Gea-Aguilera, F., Gill, J. & Zhang, X. 2017 Synthetic turbulence methods for computational aeroacoustic simulations of leading edge noise. Comput. Fluids 157, 240–252.Google Scholar
Gea-Aguilera, F., Gill, J., Zhang, X., Chen, X. X. & Nodé-Langlois, T.2016 Leading edge noise predictions using anisotropic synthetic turbulence. AIAA Paper 2016-2840.Google Scholar
Gershfeld, J. 2004 Leading edge noise from thick foils in turbulent flows. J. Acoust. Soc. Am. 116 (3), 1416–1426.Google Scholar
Gill, J., Zhang, X. & Joseph, P. 2013 Symmetric airfoil geometry effects on leading edge noise. J. Acoust. Soc. Am. 134 (4), 2669–2680.Google Scholar
Glegg, S. A. L., Baxter, S. M. & Glendinning, A. G. 1987 The prediction of broadband noise from wind turbines. J. Sound Vib. 118 (2), 217–239.Google Scholar
Glegg, S. A. L. & Devenport, W. J. 2010 Panel methods for airfoils in turbulent flow. J. Sound Vib. 329 (18), 3709–3720.Google Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (3), 433–468.Google Scholar
Goldstein, M. E. & Atassi, H. M. 1976 A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust. J. Fluid Mech. 74 (4), 741–765.Google Scholar
Goodwine, B. 2010 Engineering Differential Equations: Theory and Applications. Springer Science & Business Media.Google Scholar
Guidati, G. & Wagner, S.1999 The influence of airfoil shape on gust–airfoil interaction noise in compressible flows. AIAA Paper 99-1843.Google Scholar
Hall, A. M., Atassi, O. V., Gilson, J. & Reba, R.2011 Effects of leading-edge thickness on high-speed airfoil–turbulence interaction noise. AIAA Paper 2011-2861.Google Scholar
Hu, F. Q., Hussaini, M. Y. & Manthey, J. L. 1996 Low-dissipation and low-dispersion Runge–Kutta schemes for computational acoustics. J. Comput. Phys. 124 (1), 177–191.Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow round two-dimensional bluff bodies. J. Fluid Mech. 61 (4), 625–706.Google Scholar
Kemp, N. H. 1952 On the lift and circulation of airfoils in some unsteady-flow problems. J. Aero. Sci. 19 (10), 713–714.Google Scholar
Kerschen, E. J. & Balsa, T. F. 1981 Transformation of the equation governing disturbances of a two-dimensional compressible. AIAA J. 19 (10), 1367–1370.Google Scholar
Kerschen, E. J. & Myers, M. R. 1987 Perfect gas effects in compressible rapid distortion theory. AIAA J. 25 (3), 504–507.Google Scholar
Kraichnan, R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13 (1), 22–31.Google Scholar
Landahl, M. T. 1989 Unsteady Transonic Flow. Cambridge University Press.Google Scholar
Lighthill, M. J. 1956 Drift. J. Fluid Mech. 1 (1), 31–53.Google Scholar
Liu, W., Kim, J. W., Zhang, X., Angland, D. & Caruelle, B. 2013 Landing-gear noise prediction using high-order finite difference schemes. J. Sound Vib. 332 (14), 3517–3534.Google Scholar
Lockard, D. P. & Morris, P. J. 1998 Radiated noise from airfoils in realistic mean flows. AIAA J. 36 (6), 907–914.Google Scholar
Lysak, P. D.2011 Unsteady lift of thick airfoils in incompressible turbulent flow. PhD thesis, The Pennsylvania State University.Google Scholar
Ma, Z. K. & Zhang, X. 2009 Numerical investigation of broadband slat noise attenuation with acoustic liner treatment. AIAA J. 47 (12), 2812–2820.Google Scholar
Magliozzi, B.1991 Propeller and propfan noise. NASA Tech. Rep. 92-10599.Google Scholar
Martinez, R. & Widnall, S. E. 1980 Unified aerodynamic-acoustic theory for a thin rectangular wing encountering a gust. AIAA J. 18 (6), 636–645.Google Scholar
Miles, J. W. 1950 On the compressibility correction for subsonic unsteady flow. J. Aero. Sci. 17 (3), 181–182.Google Scholar
Miotto, R. F., Wolf, W. R. & de Santana, L. D. 2017 Numerical computation of aeroacoustic transfer functions for realistic airfoils. J. Sound Vib. 407, 253–270.Google Scholar
Miotto, R. F., Wolf, W. R. & de Santana, L. D. 2018 Leading-edge noise prediction of general airfoil profiles with spanwise-varying inflow conditions. AIAA J. 56 (5), 1711–1716.Google Scholar
Moreau, S., Roger, M. & Jurdic, V.2005 Effect of angle of attack and airfoil shape on turbulence-interaction noise. AIAA Paper 2005-2973.Google Scholar
Moriarty, P., Guidati, G. & Migliore, P.2005 Prediction of turbulent inflow and trailing-edge noise for wind turbines. AIAA Paper 2005-2881.Google Scholar
Myers, M. R. & Kerschen, E. J. 1997 Influence of camber on sound generation by airfoils interacting with high-frequency gusts. J. Fluid Mech. 353, 221–259.Google Scholar
Node-Langlois, T., Wlassow, F., Languille, V., Colin, Y., Caruelle, B., Gill, J., Chen, X. X., Zhang, X. & Parry, A. B.2014 Prediction of contra-rotating open rotor broadband noise in isolated and installed configurations. AIAA Paper 2014-2610.Google Scholar
Oerlemans, S. & Migliore, P.2004 Aeroacoustic wind tunnel tests of wind turbine airfoils. AIAA Paper 2004-3042.Google Scholar
Olsen, W. & Wagner, J. 1982 Effect of thickness on airfoil surface noise. AIAA J. 20 (3), 437–439.Google Scholar
Osborne, C. 1973 Unsteady thin-airfoil theory for subsonic flow. AIAA J. 11, 205–209.Google Scholar
Paterson, R. W. & Amiet, R. K. 1977 Noise and surface pressure response of an airfoil to incident turbulence. J. Aircraft 14 (8), 729–736.Google Scholar
Richards, S. K., Zhang, X., Chen, X. X. & Nelson, P. A. 2004 The evaluation of non-reflecting boundary conditions for duct acoustic computation. J. Sound Vib. 270 (3), 539–557.Google Scholar
Rienstra, S. W. & Hirschberg, A. 2004 An Introduction to Acoustics. p. 278. Eindhoven University of Technology.Google Scholar
Roger, M. & Moreau, S. 2010 Extensions and limitations of analytical airfoil broadband noise models. Intl J. Aeroacoust. 9 (3), 273–306.Google Scholar
Santana, L. D., Christophe, J., Schram, C. & Desmet, W. 2016 A rapid distortion theory modified turbulence spectra for semi-analytical airfoil noise prediction. J. Sound Vib. 383, 349–363.Google Scholar
Santana, L. D., Schram, C. & Desmet, W.2012 Panel method for turbulence–airfoil interaction noise prediction. AIAA Paper 2012-2073.Google Scholar
Schwarzschild, K. 1901 Die Beugung und Polarisation des Lichts durch einen Spalt. I. Math. Ann. 55 (2), 177–247.Google Scholar
Scott, J. & Atassi, H.1990 Numerical solutions of the linearized Euler equations for unsteady vortical flows around lifting airfoils. AIAA Paper 90-0694.Google Scholar
Sears, W. R.1938 A systematic presentation of the theory of thin airfoils in non-uniform motion. PhD thesis, California Institute of Technology.Google Scholar
Sears, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical application. J. Aero. Sci. 8 (3), 104–108.Google Scholar
Shen, Z. & Zhang, X.2018 Random-eddy superposition technique for leading edge noise predictions. AIAA Paper 2018-3597.Google Scholar
Staubs, J. K.2008 Real airfoil effects on leading edge noise. PhD thesis, Virginia Polytechnic Institute and State University.Google Scholar
Tsai, C. T.1992 Effect of airfoil thickness on high-frequency gust interaction noise. PhD thesis, The University of Arizona.Google Scholar
von Kármán, T. H. & Sears, W. R. 1938 Airfoil theory for non-uniform motion. J. Aero. Sci. 5 (10), 379–390.Google Scholar
Wang, X., Hu, Z. W. & Zhang, X. 2013 Aeroacoustic effects of high-lift wing slat track and cut-out system. Intl J. Aeroacoust. 12 (3), 283–308.Google Scholar
Zhang, X., Chen, X. X., Morfey, C. L. & Nelson, P. A. 2004 Computation of spinning modal radiation from an unflanged duct. AIAA J. 42 (9), 1795–1801.Google Scholar
Zhong, S. Y. & Zhang, X. 2017 A sound extrapolation method for aeroacoustics far-field prediction in presence of vortical waves. J. Fluid Mech. 820, 424–450.Google Scholar
Zhong, S. Y. & Zhang, X. 2018a A generalized sound extrapolation method for turbulent flows. Proc. R. Soc. Lond. A 474 (2210), 20170614.Google Scholar
Zhong, S. Y. & Zhang, X. 2018b On the frequency domain formulation of the generalized sound extrapolation method. J. Acoust. Soc. Am. 144 (24), 24–31.Google Scholar
Zhong, S. Y. & Zhang, X. 2019 On the effect of streamwise disturbance on the airfoil–turbulence interaction noise. J. Acoust. Soc. Am. 145 (4), 2530–2539.Google Scholar
Zhong, S. Y., Zhang, X., Gill, J. & Fattah, R.2017 A numerical investigation of the airfoil–gust interaction noise in transonic flows. AIAA Paper 2017-3369.Google Scholar
Zhong, S. Y., Zhang, X., Gill, J., Fattah, R. & Sun, Y. H. 2018 A numerical investigation of the airfoil–gust interaction noise in transonic flows: acoustic processes. J. Sound Vib. 425, 239–256.Google Scholar
Zhong, S. Y., Zhang, X., Gill, J., Fattah, R. & Sun, Y. H. 2019 Geometry effect on the airfoil–gust interaction noise in transonic flows. Aero. Sci. Tech. 92, 181–191.Google Scholar