Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T22:11:41.896Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  08 January 2020

Peter J. Baddoo Open the ORCID record for Peter J. Baddoo [Opens in a new window]  and
Lorna J. Ayton Open the ORCID record for Lorna J. Ayton [Opens in a new window]
Peter J. Baddoo*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WA, UK
Lorna J. Ayton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WA, UK
*
†Email address for correspondence: baddoo@damtp.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Acoustics: Aeroacoustics Waves/Free-surface Flows: Wave scattering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 886 , 10 March 2020 , A1
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ayton, L. J. 2017 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., Gill, J. R. & Peake, N. 2016 The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction. J. Sound Vib. 378, 28–37.CrossRefGoogle Scholar
Ayton, L. J. & Peake, N. 2013 On high-frequency noise scattering by aerofoils in flow. J. Fluid Mech. 734, 144–182.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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, 190–213.CrossRefGoogle Scholar
Dittmar, J. H.1972 Methods for reducing blade passing frequency noise generated by rotor–wake–stator interaction. Tech. Rep. NASA.Google Scholar
Erdogan, F. & Gupta, G. D. 1972 On the numerical solution of singular integral equations. Q. Appl. Maths 29 (4), 525–534.CrossRefGoogle Scholar
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, 25–52.CrossRefGoogle Scholar
Falcão, A. F. de O. 1975 Three-dimensional potential flow through a rectilinear cascade of blades. Ing.-Arch. 44 (1), 27–41.CrossRefGoogle Scholar
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. 149–176. Springer.CrossRefGoogle Scholar
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. 1–8.Google Scholar
Glegg, S. A. L. 1999 The response of a swept blade row to a three-dimensional gust. J. Sound Vib. 227 (1), 29–64.CrossRefGoogle Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (03), 433.Google Scholar
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), 131–143.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Hall, K. C. 1997 Exact solution to category 3 problems: turbomachinery noise. In Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems, pp. 41–43.Google Scholar
Hixon, R., Sescu, A. & Allampalli, V. 2010 Towards the prediction of noise from realistic rotor wake/stator interaction using CAA. Procedia IUTAM 1, 203–213.CrossRefGoogle Scholar
Kerschen, E. J. & Myers, M. R. 1987 Perfect gas effects in compressible rapid distortion theory. AIAA J. 25 (3), 504–507.CrossRefGoogle Scholar
Koch, W. 1971 On the transmission of sound waves through a blade row. J. Sound Vib. 18 (1), 111–128.CrossRefGoogle Scholar
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.CrossRefGoogle Scholar
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, 271–304.CrossRefGoogle 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.CrossRefGoogle Scholar
Peake, N. 1992 The interaction between a high-frequency gust and a blade row. J. Fluid Mech. 241, 261–289.CrossRefGoogle Scholar
Peake, N. 1993 The scattering of vorticity waves by an infinite cascade of flat plates in subsonic flow. Wave Motion 18 (3), 255–271.CrossRefGoogle Scholar
Peake, N. & Kerschen, E. J. 1995 A uniform asymptotic approximation for high-frequency unsteady cascade flow. Proc. R. Soc. Lond. A 449 (1935), 177–186.CrossRefGoogle Scholar
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, 315–346.CrossRefGoogle Scholar
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, 315–346.CrossRefGoogle Scholar
Peake, N. & Parry, A. B. 2012 Modern challenges facing turbomachinery aeroacoustics. Annu. Rev. Fluid Mech. 44 (1), 227–248.CrossRefGoogle Scholar
Posson, H., Roger, M. & Moreau, S. 2010 On a uniformly valid analytical rectilinear cascade response function. J. Fluid Mech. 663, 22–52.Google Scholar
Rienstra, S. W. 1992 A note on the Kutta condition in Glauert’s solution of the thin aerofoil problem. J. Engng Maths 26 (1), 61–69.CrossRefGoogle Scholar
Robinson, A. & Laurmann, J. A. 1956 Wing Theory. Cambridge University Press.Google Scholar
Schulten, J. B. H. M. 1982 Sound generated by rotor wakes interacting with a leaned vane stator. AIAA J. 20 (10), 1352–1358.CrossRefGoogle Scholar
Smith, S. N.1973 Discrete frequency sound generation in axial flow turbomachines. Aeronaut. Res. Counc. R&M 3709, pp. 1–58.Google Scholar
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.CrossRefGoogle Scholar
Tsai, C.-T.1992 Effect of airfoil thickness on high-frequency gust interaction noise. PhD thesis, The University of Arizona.Google Scholar
Verdon, J. M. 1993 Review of unsteady aerodynamic methods for turbomachinery aeroelastic and aeroacoustic applications. AIAA J. 31 (2), 235–250.CrossRefGoogle Scholar
Supplementary material: File

Baddoo and Ayton supplementary material

Baddoo and Ayton supplementary material

Download Baddoo and Ayton supplementary material(File)
File 794.7 KB