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Temporal instability modes of supersonic round jets

Published online by Cambridge University Press:  01 September 2010

LUIS PARRAS  and
STÉPHANE LE DIZÈS
LUIS PARRAS*
Affiliation:
IRPHE-CNRS & Aix-Marseille University, 49 rue F. Joliot-Curie, F-13013 Marseille, France Universidad de Málaga, E. T. S. Ingenieros Industriales, 29071 Málaga, Spain
STÉPHANE LE DIZÈS
Affiliation:
IRPHE-CNRS & Aix-Marseille University, 49 rue F. Joliot-Curie, F-13013 Marseille, France
*
†Email address for correspondence: [email protected]
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Abstract

In this study, a comprehensive inviscid temporal stability analysis of a compressible round jet is performed for Mach numbers ranging from 1 to 10. We show that in addition to the Kelvin–Helmholtz instability modes, there exist for each azimuthal wavenumber three other types of modes (counterflow subsonic waves, subsonic waves and supersonic waves) whose characteristics are analysed in detail using a WKBJ theory in the limit of large axial wavenumber. The theory is constructed for any velocity and temperature profile. It provides the phase velocity and the spatial structure of the modes and describes qualitatively the effects of base-flow modifications on the mode characteristics. The theoretical predictions are compared with numerical results obtained for an hyperbolic tangent model and a good agreement is demonstrated. The results are also discussed in the context of jet noise. We show how the theory can be used to determine a priori the impact of jet modifications on the noise induced by instability.

JFM classification

Acoustics: Jet noise Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 662 , 10 November 2010 , pp. 173 - 196
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.CrossRefGoogle Scholar
Blumen, W. 1970 Shear layer instability of an inviscid compressible fluid. J. Fluid Mech. 40, 769–781.CrossRefGoogle Scholar
Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear layer instability of an inviscid compressible fluid. Part 2. J. Fluid Mech. 71, 305–316.CrossRefGoogle Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.CrossRefGoogle Scholar
Duck, P. 1990 The inviscid axisymmetric stability of the supersonic flow along a circular cylinder. J. Fluid Mech. 214, 661–637.CrossRefGoogle Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl numbers. J. Fluid Mech. 500, 239–262.CrossRefGoogle Scholar
Landau, L. 1944 Stability of tangential discontinuities in compressible fluid. Akad. Nauk. S.S.S.R., Comptes Rendus (Doklady) 44, 139–141.Google Scholar
Lau, J. C. 1981 Effects of exit mach number an temperature on mean flow and turbulence characteristics in round jets. J. Fluid Mech. 105, 193–218.CrossRefGoogle Scholar
Le Dizès, S. 2008 Inviscid waves on a Lamb–Oseen vortex in a rotating stratified fluid: consequences on the elliptic instability. J. Fluid Mech. 597, 283–303.CrossRefGoogle Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.CrossRefGoogle Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex kelvin modes. J. Fluid Mech. 542, 69–96.CrossRefGoogle Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211, 564–587.Google Scholar
Lindzen, R. & Barker, J. W. 1985 Instability and wave over-reflection in stably stratified shear flow. J. Fluid Mech. 151, 189–217.CrossRefGoogle Scholar
Luo, K. H. & Sandham, N. D. 1996 Instability of vortical and acoustic modes in supersonic round jets. Phys. Fluids 9, 1003–1013.CrossRefGoogle Scholar
Mack, L. M. 1984 Boundary layer linear stability theory. AGARD Rep. 709.Google Scholar
Mack, L. M. 1990 On the inviscid acoustic-mode instability of supersonic shear flows. Part I. Two-dimensional waves. Theor. Comput. Fluid Dyn. 2, 97.CrossRefGoogle Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent profile. J Fluid Mech. 19, 543–556.CrossRefGoogle Scholar
Papamoschou, D. & Debiasi, M. 2001 Directional suppression of noise from a high-speed jet. AIAA 39 (3), 380–387.CrossRefGoogle Scholar
Riedinger, X., Le Dizès, S. & Meunier, P. 2010 Viscous stability properties of Lamb–Oseen vortex in a stratified fluid. J. Fluid Mech. 645, 255–278.CrossRefGoogle Scholar
Shepard, H. K. 1983 Decay widths for metastable states. Improved WKB approximation. Phys. Rev. D 27 (6), 1288–1298.CrossRefGoogle Scholar
Takehiro, S. & Hayashi, Y. Y. 1992 Over-reflection and shear instability in a shallow-water model. J. Fluid Mech. 236, 259–279.CrossRefGoogle Scholar
Tam, C. W. 1995 Supersonic jet noise. Annu. Rev. Fluid Mech. 27, 17–43.CrossRefGoogle Scholar
Tam, C. K. W. & Burton, D. E. 1984 a Sound generated by instability waves of supersonic flows. Part 1. Two-dimensional mixing layers. J. Fluid Mech. 138, 249–271.CrossRefGoogle Scholar
Tam, C. K. W. & Burton, D. E. 1984 b Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets. J. Fluid Mech. 138, 273–295.CrossRefGoogle Scholar
Tam, C. K. W. & Hu, F. Q. 1989 On the three families of instability waves of high speed jets. J. Fluid Mech. 201, 447–483.CrossRefGoogle Scholar
Troutt, T. R. & McLaughlin, D. K. 1982 Experiments of the flow and acoustic properties of a moderate-Reynolds-number supersonic jet. J. Fluid Mech. 116, 123–156.CrossRefGoogle Scholar