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An analytic solution for gust–cascade interaction noise including effects of realistic aerofoil geometry

  • Peter J. Baddoo (a1) and Lorna J. Ayton (a1)


This paper presents an analytic solution for the sound generated by rotor–stator interaction for aerofoils with small camber and thickness subject to a background flow with small angle of attack. The interaction is modelled as a convected, unsteady vortical or entropic gust incident on an infinite rectilinear cascade of staggered aerofoils in a background flow that is uniform far away from the cascade. Applying rapid distortion theory (RDT) and transforming to an orthogonal coordinate system reduces the cascade of aerofoils to a cascade of flat plates. By seeking a perturbation expansion in terms of the disturbance of the background flow from uniform flow, leading- and first-order governing equations and boundary conditions are obtained for the acoustic potential. The system is then solved analytically using the Wiener–Hopf method. The resulting expression is inverted to give the acoustic potential function in the entire domain, i.e. a solution to the inhomogeneous convected Helmholtz equation with inhomogeneous boundary conditions in a cascade geometry. The solution significantly extends previous analytical work that is restricted to flat plates or only calculates the far-upstream radiation, and as such can give insight into the role played by blade geometry on the acoustic field upstream, downstream and in the important inter-blade region of the cascade. This new solution is validated against solutions that only account for flat plates at zero angle of attack. Various aeroacoustic results, including the scattered pressure, unsteady lift and sound power output, are discussed for a range of geometries and angles of attack.


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Ayton, L. J. 2017 An analytic solution for gust–aerofoil interaction noise including effects of geometry. IMA J. Appl. Maths 82 (2), 280304.
Ayton, L. J., Gill, J. R. & Peake, N. 2016 The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction. J. Sound Vib. 378, 2837.
Ayton, L. J. & Peake, N. 2013 On high-frequency noise scattering by aerofoils in flow. J. Fluid Mech. 734, 144182.
Baddoo, P. J. & Ayton, L. J. 2017 An analytic solution for gust–cascade interaction noise including effects of realistic aerofoil geometry. In 23rd AIAA/CEAS Aeroacoustics Conf., Reston, Virginia, AIAA Paper 2017-3216.
Baddoo, P. J. & Ayton, L. J. 2018a An analytic solution for gust–cascade interaction noise including effects of realistic aerofoil geometry: inter-blade region. In 2018 AIAA/CEAS Aeroacoustics Conf. AIAA Aviat. Forum, Atlanta, Georgia, AIAA Paper 2018-2957.
Baddoo, P. J. & Ayton, L. J. 2018b Potential flow through a cascade of aerofoils: direct and inverse problems. Proc. R. Soc. Lond. A 474 (2217), 20180065.
Bouley, S., François, B., Roger, M., Posson, H. & Moreau, S. 2017 On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes. J. Sound Vib. 403, 190213.
Dittmar, J. H.1972 Methods for reducing blade passing frequency noise generated by rotor–wake–stator interaction. Tech. Rep. NASA.
Erdogan, F. & Gupta, G. D. 1972 On the numerical solution of singular integral equations. Q. Appl. Maths 29 (4), 525534.
Evers, I. & Peake, N. 2002 On sound generation by the interaction between turbulence and a cascade of airfoils with non-uniform mean flow. J. Fluid Mech. 463, 2552.
Falcão, A. F. de O. 1975 Three-dimensional potential flow through a rectilinear cascade of blades. Ing.-Arch. 44 (1), 2741.
Fang, J. & Atassi, H. M. 1993 Compressible flows with vortical disturbances around a cascade of loaded airfoils. In Unsteady Aerodynamics Aeroacoustics, Aeroelasticity Turbomachines Propellers, pp. 149176. Springer.
Gea-Aguilera, F., Gill, J., Zhang, X. & Nodé-Langlois, T. 2016 Turbulence–cascade interaction noise using an advanced digital filter method. In 23rd International Congress of Sound and Vibration, pp. 18.
Glegg, S. A. L. 1999 The response of a swept blade row to a three-dimensional gust. J. Sound Vib. 227 (1), 2964.
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (03), 433.
Grace, S. M. 2016 Influence of model parameters and the vane response method on a low-order prediction of fan broadband noise. Intl J. Aeroacoust. 15 (1–2), 131143.
Guzman Inigo, J., Baddoo, P. J., Morgans, A. S. & Ayton, L. J. 2019 Noise generated by entropic and compositional inhomogeneities interacting with a cascade of airfoils. In AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2019-2526.
Guzman Inigo, J., Duran, I. & Morgans, A. 2018 A model for the sound generated by entropy disturbances interacting with isolated blades. In 2018 AIAA/CEAS Aeroacoustics Conference, American Institute of Aeronautics and Astronautics.
Hall, K. C. 1997 Exact solution to category 3 problems: turbomachinery noise. In Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems, pp. 4143.
Hixon, R., Sescu, A. & Allampalli, V. 2010 Towards the prediction of noise from realistic rotor wake/stator interaction using CAA. Procedia IUTAM 1, 203213.
Kerschen, E. J. & Myers, M. R. 1987 Perfect gas effects in compressible rapid distortion theory. AIAA J. 25 (3), 504507.
Koch, W. 1971 On the transmission of sound waves through a blade row. J. Sound Vib. 18 (1), 111128.
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Myers, M. R. & Kerschen, E. J. 1995 Influence of incidence angle on sound generation by airfoils interacting with high-frequency gusts. J. Fluid Mech. 292, 271304.
Myers, M. R. & Kerschen, E. J. 1997 Influence of camber on sound generation by airfoils interacting with high-frequency gusts. J. Fluid Mech. 353, 221259.
Peake, N. 1992 The interaction between a high-frequency gust and a blade row. J. Fluid Mech. 241, 261289.
Peake, N. 1993 The scattering of vorticity waves by an infinite cascade of flat plates in subsonic flow. Wave Motion 18 (3), 255271.
Peake, N. & Kerschen, E. J. 1995 A uniform asymptotic approximation for high-frequency unsteady cascade flow. Proc. R. Soc. Lond. A 449 (1935), 177186.
Peake, N. & Kerschen, E. J. 1997 Influence of mean loading on noise generated by the interaction of gusts with a flat-plate cascade: upstream radiation. J. Fluid Mech. 347, 315346.
Peake, N. & Kerschen, E. J. 2004 Influence of mean loading on noise generated by the interaction of gusts with a cascade: downstream radiation. J. Fluid Mech. 515, 315346.
Peake, N. & Parry, A. B. 2012 Modern challenges facing turbomachinery aeroacoustics. Annu. Rev. Fluid Mech. 44 (1), 227248.
Posson, H., Roger, M. & Moreau, S. 2010 On a uniformly valid analytical rectilinear cascade response function. J. Fluid Mech. 663, 2252.
Rienstra, S. W. 1992 A note on the Kutta condition in Glauert’s solution of the thin aerofoil problem. J. Engng Maths 26 (1), 6169.
Robinson, A. & Laurmann, J. A. 1956 Wing Theory. Cambridge University Press.
Schulten, J. B. H. M. 1982 Sound generated by rotor wakes interacting with a leaned vane stator. AIAA J. 20 (10), 13521358.
Smith, S. N.1973 Discrete frequency sound generation in axial flow turbomachines. Aeronaut. Res. Counc. R&M 3709, pp. 1–58.
Thwaites, B. 1960 Incompressible Aerodynamics: An Account of the Theory and Observation of the Steady Flow of Incompressible Fluid Past Aerofoils, Wings, and Other Bodies. Dover Publications.
Tsai, C.-T.1992 Effect of airfoil thickness on high-frequency gust interaction noise. PhD thesis, The University of Arizona.
Verdon, J. M. 1993 Review of unsteady aerodynamic methods for turbomachinery aeroelastic and aeroacoustic applications. AIAA J. 31 (2), 235250.
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An analytic solution for gust–cascade interaction noise including effects of realistic aerofoil geometry

  • Peter J. Baddoo (a1) and Lorna J. Ayton (a1)


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