Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T16:11:22.486Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolute and convective instabilities in free shear layers

Published online by Cambridge University Press:  20 April 2006

P. Huerre  and
P. A. Monkewitz
P. Huerre
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90089–0192
P. A. Monkewitz
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, University of California at Los Angeles, Los Angeles, California 90024
Get access Rights & Permissions [Opens in a new window]

Abstract

The absolute or convective character of inviscid instabilities in parallel shear flows can be determined by examining the branch-point singularities of the dispersion relation for complex frequencies and wavenumbers. According to a criterion developed in the study of plasma instabilities, a flow is convectively unstable when the branch-point singularities are in the lower half complex-frequency plane. These concepts are applied to a family of free shear layers with varying velocity ratio $R = \Delta U/2\overline{U}$, where ΔU is the velocity difference between the two streams and $\overline{U}$ their average velocity. It is demonstrated that spatially growing waves can only be observed if the mixing layer is convectively unstable, i.e. when the velocity ratio is smaller than Rt = 1.315. When the velocity ratio is larger than Rt, the instability develops temporally. Finally, the implications of these concepts are discussed also for wakes and hot jets.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 159 , October 1985 , pp. 151 - 168
Copyright
© 1985 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

Bechert, D. & Pfitzenmaier, E. 1975 On wavelike perturbations in a free jet travelling faster than the mean flow in the jet. J. Fluid Mech. 72, 341–352.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Bers, A. 1975 Linear waves and instabilities. In Physique des Plasmas (ed. C. DeWitt & J. Peyraud), pp. 117–213. New York: Gordon and Breach.
Betchov, R. & Criminale, W. O. 1966 Spatial instability of the inviscid jet and wake. Phys. Fluids 9, 359–362.Google Scholar
Briggs, R. J. 1964 Electron-stream interaction with plasmas. Research Monograph No. 29. Cambridge, Mass: M.I.T Press.
Browand, F. K. 1966 An experimental investigation of the instability of an incompressible, separated shear layer. J. Fluid Mech. 26, 281–307.Google Scholar
Brown, S. N. & Stewartson, K. 1980 On the algebraic decay of disturbances in a stratified shear flow. J. Fluid Mech. 100, 811–816.Google Scholar
Case, K. M. 1960 Stability of inviscid Couette flow. Phys. Fluids 3, 143–148.Google Scholar
Chimonas, G. 1979 Algebraic disturbances in stratified shear flows. J. Fluid Mech. 90, 1–19.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Freymuth, P. 1966 On transition in a separated laminar boundary layer. J. Fluid Mech. 25, 683–704.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.Google Scholar
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11, 723–727.Google Scholar
Gaster, M. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347, 271–289.Google Scholar
Gaster, M. 1980 The propagation of wave packets in laminar boundary layers: asymptotic theory for non-conservative wave systems. Preprint, National Maritime Institute, Teddington, England.
Ho, C. M. & Huang, L. S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Koch, W. 1983 Organized structures in wakes and jets – An aerodynamic resonance phenomenon? Proc. 4th Symp. on Turb. shear Flows at Karlsruhe. Springer.
Koch, W. 1985 Local instability characteristics and frequency determination of self-excited wake flows. J. Sound and Vib. 99, in press.Google Scholar
Lighthill, J. 1981 Energy flow in the cochlea. J. Fluid Mech. 106, 149–213.Google Scholar
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51, 233–272.Google Scholar
Merkine, L. O. 1977 Convective and absolute instability of baroclinic eddies. Geophys. Astrophys. Fluid Dyn. 9, 129–157.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic tangent velocity profile. J. Fluid Mech. 19, 543–556.Google Scholar
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521–544.Google Scholar
Michalke, A. 1970 A note on the spatial jet-instability of the compressible cylindrical vortex sheet. DLR research rep. FB-70-51.
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistrahls unter Berücksichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Flugwiss. 19, 319–328.Google Scholar
Miksad, R. W. 1972 Experiments on the nonlinear stages of free shear layer transition. J. Fluid Mech. 56, 695–719.Google Scholar
Monkewitz, P. A. & Huerre, P. 1982 Influence of the velocity ratio on the spatial instability of mixing layers. Phys. Fluids 25, 1137–1143.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
Tatsumi, T., Gotoh, K. & Ayukawa, K. 1964 The stability of a free boundary layer at large Reynolds numbers. J. Phys. Soc. Japan 19, 1966–1980.Google Scholar
Thorpe, S. A. 1971 Experiments on the instability of stratified shear flows: immiscible fluids. J. Fluid Mech. 46, 299–319.Google Scholar
Weissman, M. A. 1979 Nonlinear wave packets in the Kelvin–Helmholtz instability. Phil. Trans. R. Soc. Lond. A 290, 639–681.Google Scholar