Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T03:21:27.168Z Has data issue: false hasContentIssue false
  • English
  • Français

The radiation of sound by the instability waves of a compressible plane turbulent shear layer

Published online by Cambridge University Press:  19 April 2006

Christopher K. W. Tam  and
Philip J. Morris
Christopher K. W. Tam
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306
Philip J. Morris
Affiliation:
Department of Aerospace Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of acoustic radiation generated by instability waves of a compressible plane turbulent shear layer is solved. The solution provided is valid up to the acoustic far-field region. It represents a significant improvement over the solution obtained by classical hydrodynamic-stability theory which is essentially a local solution with the acoustic radiation suppressed. The basic instability-wave solution which is valid in the shear layer and the near-field region is constructed in terms of an asymptotic expansion using the method of multiple scales. This solution accounts for the effects of the slightly divergent mean flow. It is shown that the multiple-scales asymptotic expansion is not uniformly valid far from the shear layer. Continuation of this solution into the entire upper half-plane is described. The extended solution enables the near-and far-field pressure fluctuations associated with the instability wave to be determined. Numerical results show that the directivity pattern of acoustic radiation into the stationary medium peaks at 20 degrees to the axis of the shear layer in the downstream direction for supersonic flows. This agrees qualitatively with the observed noise-directivity patterns of supersonic jets.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 2 , 29 May 1980 , pp. 349 - 381
Copyright
© 1980 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

Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Bishop, K. A., Ffowcs Williams, J. E. & Smith, W. 1971 On the noise sources of the un-suppressed high speed jet. J. Fluid Mech. 50, 21–31.Google Scholar
Blumen, W. 1970 Shear layer instability of an inviscid compressible fluid. J. Fluid Mech. 40, 769–781.Google Scholar
Blumen, W. 1971 Jet flow instability of an inviscid compressible fluid. J. Fluid Mech. 46, 737–747.Google Scholar
Bouthier, M. 1972 Stabilité linéaire des écoulements presque parallèles. Partie I. J. Méc. 11, 599–621.Google Scholar
Bouthier, M. 1973 Stabilité linéaire des écoulements presque parallèles. Partie II. La Couche Limite de Blasius. J. Méc. 12, 75–95.Google Scholar
Boyce, W. E. & DiPrima, R. C. 1977 Elementary Differential Equations, 3rd edn. Wiley.
Brigham, E. O. & Morrow, R. E. 1967 The fast Fourier transform. I.E.E.E. Spectrum, December, pp. 63–70.Google Scholar
Chan, Y. Y. 1974a Spatial waves in turbulent jets. Phys. Fluids 17, 46–53.Google Scholar
Chan, Y. Y. 1974b Spatial waves in turbulent jets. Part II. Phys. Fluids 17, 1667–1670.Google Scholar
Chan, Y. Y. 1975 Nonlinear spatial wave development in an axisymmetric turbulent jet. Nat. Res. Counc., Canada, Nat. Aero. Est. Aero. Rep. LR-585.Google Scholar
Chan, Y. Y. 1976 Spatial waves of higher order modes in an axisymmetric turbulent jet. Phys. Fluids 19, 2042–2043.Google Scholar
Crighton, D. B. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413.Google Scholar
Dahan, C. & Élias, G. 1976 Source structure pattern in a hot jet by infrared-microphones correlations. A.I.A.A. Paper 76–542.Google Scholar
Dikii, L. A. 1960 The stability of plane parallel flows of an ideal fluid. Sov. Phys. Dokl. 135, 1179–1182.Google Scholar
Dosanjh, D. S. & Yu, J. C. 1968 Noise from underexpanded axisymmetric jet flow using radial jet flow impingement. Proc. AFOSR-UTIAS Symp. Aerodyn. Noise, Toronto, Canada.Google Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480.Google Scholar
Gropengieser, H. 1969 Bietrag zur Stabilität freier Grenzschichten in kompressiblen Medien. Deutsch Luft-und Raumfahrt FB 69–25, Berlin.Google Scholar
Lau, J. C., Morris, P. J. & Fisher, M. J. 1976 Turbulence measurements in subsonic and supersonic jets using a laser velocimeter. A.I.A.A. Paper no. 76, 348.Google Scholar
Lees, L. & Lin, C. C. 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. N.A.C.A. Tech. note 1115.Google Scholar
Liepmann, H. W. & Laufer, J. 1947 Investigations of free turbulent mixing. N.A.C.A. Tech. Note 1257.Google Scholar
Lin, C. C. 1953 On the stability of laminar mixing region between two parallel streams in a gas. N.A.C.A. Tech. Note 2887.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Liu, J. T. C. 1974 Developing large-scale wave-like eddies and the near-jet noise field. J. Fluid Mech. 62, 437–464.Google Scholar
Mack, L. M. 1965 Computation of the stability of the laminar compressible boundary layer. Methods in Comp. Phys. 4, 147–299.Google Scholar
McLaughlin, D. K., Morrison, G. L. & Troutt, T. R. 1975 Experiments on the instability waves in a supersonic jet and their acoustic radiation. J. Fluid Mech. 69, 73–95.Google Scholar
McLaughlin, D. K., Morrison, G. L. & Troutt, T. R. 1977 Reynolds number dependence in supersonic jet noise. A.I.A.A. J. 15, 526–532.Google Scholar
Merkine, L. & Liu, J. T. C. 1975 On the development of noise-producing large-scale wave-like eddies in a plane turbulent jet. J. Fluid Mech. 70, 353–368.Google Scholar
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistrahls unter Berücksichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Flugwiss. 19, 319–328.Google Scholar
Moore, C. J. 1977 The role of shear layer instability waves in jet exhaust noise. J. Fluid Mech. 80, 321–367.Google Scholar
Morris, P. J. 1974 A model for the structure of jet turbulence as a source of noise. A.I.A.A. Paper 74, 1.Google Scholar
Morris, P. J. 1976a The flow and acoustic characteristics of the large-scale wave-like structure of an axisymmetric jet. In The Generation and Radiation of Supersonic Jet Noise (ed. H. E. Plumblee), vol. 2, ch. 4 (AFAPL-TR-76-65).
Morris, P. J. 1976b The spatial viscous instability of axisymmetric jets. J. Fluid Mech. 77, 511–529.Google Scholar
Morris, P. J. 1977 Flow characteristics of the large-scale wave-like structure of a supersonic round jet. J. Sound Vib. 53, 223–244.Google Scholar
Patel, R. P. 1973 An experimental study of a plane mixing layer. A.I.A.A. J. 11, 67–71.Google Scholar
Saric, W. S. & Nayfeh, A. H. 1975 Nonparallel stability of boundary layer flows. Phys. Fluids 18, 945–950.Google Scholar
Sedel'nikov, T. K. 1967 The frequency spectrum of the noise of a supersonic jet. Phys. Aero. Noise Moscow: Nauka. (Trans. 1969 NASA TTF-538, pp. 71–75.)Google Scholar
Tam, C. K. W. 1971 Directional acoustic radiation from a supersonic jet generated by shear layer instability. J. Fluid Mech. 46, 757–768.Google Scholar
Tam, C. K. W. 1972 On the noise of a nearly ideally expanded supersonic jet. J. Fluid Mech. 51, 69–95.Google Scholar
Tam, C. K. W. 1975 Supersonic jet noise generated by large-scale disturbances. J. Sound Vib. 38, 51–79.Google Scholar
Tam, C. K. W. 1978 Excitation of instability waves in a two-dimensional shear layer by sound. J. Fluid Mech. 89, 357–371.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Stanford, California: Parabolic.