Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T15:23:45.020Z Has data issue: false hasContentIssue false
  • English
  • Français

The interaction between a high-frequency gust and a blade row

Published online by Cambridge University Press:  26 April 2006

N. Peake
N. Peake
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The ingestion of convected vorticity by a high-solidity rotating blade row is a potent noise source in modern aeroengines, due largely to the high level of mutual aerodynamic interactions between adjacent blades. In order to model this process we solve the problem of determining the unsteady lift on an infinite cascade of finite-chord flat plates due to an incident vorticity wave. The method of solution is the Wiener–Hopf technique, and we consider the case of the reduced frequency, Ω, being large, allowing application of asymptotic analysis in the formal limit Ω → ∞. This approach yields considerable simplification, both in allowing the truncation of an infinite reflection series to just two terms, and in allowing algebraic expressions for the Wiener–Hopf split functions to be found. The unsteady lift distribution is derived in closed form, and the accuracy of the asymptotic Wiener–Hopf factorization demonstrated for even modest values of Ω by comparison with exact (but less tractable) methods. Our formulae can easily be incorporated into existing noise prediction codes: the advantage of our scheme is that it handles a regime in which conventional numerical approaches become unwieldy, as well as providing significant physical insight into the underlying mechanisms.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 261 - 289
Copyright
© 1992 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

Abramowitz, M. & Stegun I. A. 1968 Handbook of Mathematical Functions. Dover.
Adamczyk, J. J. & Goldstein M. E. 1978 Unsteady flow in a supersonic cascade with subsonic leading-edge locus. AIAA J. 16 1248–1254.Google Scholar
Cargill A. M. 1988 On high frequency cascade/gust interaction. Rolls-Royce Rep. TSG 0419.Google Scholar
Carlson, J. F. & Heins A. E. 1946 The reflection of electromagnetic waves by an infinite set of plates, I. Q. Appl. Maths. 4, 313–329.Google Scholar
Crighton D. G. 1971 Acoustic beaming and reflexion from wave-bearing surfaces. J. Fluid Mech. 47, 625–638.Google Scholar
Crighton D. G. 1985 The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech. 17, 411–445.Google Scholar
Goldstein M. E. 1976 Aeroacoustics. McGraw-Hill.
Heins A. E. 1950 The reflection of electromagnetic waves by an infinite set of plates, III. Q. Appl. Maths. 8, 281–291.Google Scholar
Heins, A. E. & Carlson J. F. 1947 The reflection of electromagnetic waves by an infinite set of plates, II. Q. Appl. Maths. 5, 82–88.Google Scholar
Jones D. S. 1986 Acoustic and Electromagnetic Waves. Oxford University Press.
Kaji, S. & Okazaki T. 1970a Propagation of sound waves through a blade row I. Analysis based on the semi-actuator disk theory. J. Sound Vib. 11, 339–353.Google Scholar
Kaji, S. & Okazaki T. 1970b Propagation of sound waves through a blade row II. Analysis based on the acceleration potential method. J. Sound Vib. 11, 355–375.Google Scholar
Koch W. 1971 On the transmission of sound waves through a blade row. J. Sound Vib. 18, 111–128.Google Scholar
Koch W. 1983 Resonant acoustic frequencies of flat plate cascades. J. Sound Vib. 88, 233–242.Google Scholar
Koiter W. T. 1954 Approximate solution of Wiener–Hopf type integral equations with applications, I–III Koninkl. Ned. Akad. Wetenschap. Proc. B 57, 558–579.Google Scholar
Landahl M. 1989 Unsteady Transonic Flow. Cambridge University Press.
Lighthill M. J. 1952 On sound generated aerodynamically. I. General theory Proc. R. Soc. Lond. A 211, 564–587.Google Scholar
Lighthill M. J. 1954 On sound generated aerodynamically. II. Turbulence as a source of sound Proc. R. Soc. Lond. A 222, 1–32.Google Scholar
Lighthill M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Mani, R. & Horvay G. 1970 Sound transmission through blade rows. J. Sound Vib. 12, 59–83.Google Scholar
Meister E. 1962a Zum Dirichlet-Problem der Helmholtzschen Schwingungsgleichung fur ein gestaffeltes Streckengitter. Arch. Rat. Mech. Anal. 10, 67–100.Google Scholar
Meister E. 1962b Zum Neumann-Problem der Helmholtzschen Schwingungsgleichung fur ein gestaffeltes Streckengitter. Arch. Rat. Mech. Anal. 10, 127–148.Google Scholar
Noble B. 1958 Methods based on the Wiener–Hopf technique. Pergamon.
Parry, A. B. & Crighton D. G. 1989 Prediction of counter-rotation propeller noise. AIAA Paper 89–1141.Google Scholar
Schwarzschild K. 1901 Die beugung und polarisation des lichts durch einen spalt – I. Math. Ann. 55, 177–247.Google Scholar
Verdon, J. M. & Hall K. C. 1990 Development of a linearised unsteady aerodynamic analysis for cascade gust response predictions. NASA Rep. CR-4308.Google Scholar
Whitehead D. S. 1970 Vibration and sound generation in a cascade of flat plates in subsonic flow. University of Cambridge Department of Engineering Rep. CUED/A-Turbo/Tr 15.Google Scholar