Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-16T22:16:16.180Z Has data issue: false hasContentIssue false
  • English
  • Français

The role of transverse secondary instabilities in the evolution of free shear layers

Published online by Cambridge University Press:  26 April 2006

G. P. Klaassen  and
W. R. Peltier
G. P. Klaassen
Affiliation:
Department of Earth and Atmospheric Science, York University, North York, Ontario, Canada M3J 1P3
W. R. Peltier
Affiliation:
Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
Get access Rights & Permissions [Opens in a new window]

Abstract

Linear stability analyses and nonlinear flow simulations reveal several important features of transverse secondary instabilities of two-dimensional Kelvin–Helmholtz billows and Stuart vortices. Vortex pairing is found to be the most rapidly amplified mode in a continuous spectrum of vortex merging instabilities. In certain not uncommon circumstances it is possible for more than two vortices to amalgamate in a single interaction, demonstrating that the phenomenon that has become known as the pairing resonance in fact has a rather low quality factor. Another form of merging instability in which a vortex is deformed and drained by its neighbours has been revealed by our linear stability analyses of nonlinear shear-layer disturbances. It appears, however, that this vortex draining instability may be important only in unstratified or very weakly stratified flows, since in moderately stratified Kelvin–Helmholtz flow, it is replaced by a highly localized instability which leads to a temporary distortion of the braids. Nonlinear simulations of vortex merging events in moderately stratified, high-Reynolds-number shear layers are compared to the theoretical predictions of our stability analyses. We investigate and quantify the sensitivity of merging events to variations in the initial conditions. The character of the flow after merging instability saturates and the nonlinear aspects of multiple merging events are also considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 202 , May 1989 , pp. 367 - 402
Copyright
© 1989 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

Bernal, L. P., Breidenthal, R. E., Brown, G. L., Konrad, J. H. & Roshko, A., 1980 On the development of three-dimensional small scales in turbulent mixing layers. In Turbulent Shear Flows 2 (ed. L. J. S. Bradbury, F. Durst, B. E. Launder, F. W. Schmidt & J. H. Whitelaw), p. 305. Springer.
Bernal, L. P. & Roshko, A., 1986 Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499–525.Google Scholar
Breidenthal, R. E.: 1981 Structure in turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1–24.Google Scholar
Browand, F. K.: 1966 An experimental investigation of the instability of an incompressible separated shear layer. J. Fluid Mech. 26, 281–307.Google Scholar
Browand, F. K. & Winant, C. D., 1973 Laboratory observations of shear-layer instability in a stratified fluid. Boundary-Layer Met. 5, 67–77.Google Scholar
Corcos, G. M. & Lin, J. S., 1984 The mixing layer: deterministic models of a turbulent flow. Part 2. The origin of the three-dimensional motion. J. Fluid Mech. 139, 67–95.Google Scholar
Corcos, G. M. & Sherman, F. S., 1976 Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73, 241–264.Google Scholar
Corcos, G. M. & Sherman, F. S., 1984 The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow. J. Fluid Mech. 139, 29–65.Google Scholar
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier–Stokes equations. Proc. R. Soc. Lond. A 406, 13–26.Google Scholar
Davis, P. A. & Peltier, W. R., 1979 Some characteristics of the Kelvin–Helmholtz and resonant overreflection modes of shear flow instability and of their interaction through vortex pairing. J. Atmos. Sci. 36, 2395.Google Scholar
Freymuth, P.: 1966 On transition in a separated laminar boundary layer. J. Fluid Mech. 25, 683–704.Google Scholar
Hernan, M. A. & Jimenez, J., 1982 Computer analysis of a high speed film of the plane turbulent mixing layer. J. Fluid Mech. 119, 323–345.Google Scholar
Ho, C.-M. & Huang, L. S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Ho, C.-M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Jimenez, J., Cogollos, M. & Bernal, L. P., 1985 A perspective view of the plane mixing layer. J. Fluid Mech. 152, 125–143.Google Scholar
Kelly, R. E.: 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657–689.Google Scholar
Klaassen, G. P. & Peltier, W. R., 1985a The evolution of finite amplitude Kelvin–Helmholtz billows in two spatial dimensions. J. Atmos. Sci. 42, 1321–1339.Google Scholar
Klaassen, G. P. & Peltier, W. R., 1985b The onset of turbulence in finite-amplitude Kelvin–Helmholtz billows. J. Fluid Mech. 155, 1–35.Google Scholar
Klaassen, G. P. & Peltier, W. R., 1985c The effect of Prandtl number on the evolution and stability of finite amplitude Kelvin–Helmholtz billows. Geophys. Astrophys. Fluid Dyn. 32, 23–60.Google Scholar
Klaassen, G. P. & Peltier, W. R., 1987 Secondary instability and transition in finite amplitude Kelvin–Helmholtz billows. In Proc. Third Intl Symp. on Stratified Flows, 3–5 Feb. 1987. Pasadena, California, vol. I (ed. E. J. List & G. Jirka).
Koop, C. G. & Browand, F. K., 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135–159.Google Scholar
Lamb, H.: 1932 Hydrodynamics, 6th edn. Dover.
Lasheras, J. C. & Choi, H., 1988 Three-dimensional instability of a plane, free shear layer: an experimental study of the formation and evolution of streamwise vortices. J. Fluid Mech. 189, 53–86.Google Scholar
Maslowe, S. A.: 1973 Finite amplitude Kelvin–Helmholtz billows. Boundary-Layer Met. 5, 43–52.Google Scholar
Miksad, R. W.: 1972 Experiments on the nonlinear stages of free-shear-layer transitions. J. Fluid Mech. 56, 695–719.Google Scholar
Nagata, M. & Busse, F. H., 1983 Three-dimensional tertiary motions in a plane shear layer. J. Fluid Mech. 135, 1–26.Google Scholar
Patnaik, P. C., Sherman, F. S. & Corcos, G. M., 1976 A numerical simulation of Kelvin–Helmholtz waves of finite amplitude. J. Fluid Mech. 73, 215–240.Google Scholar
Peltier, W. R., Hallé, J. & Clark, T. L. 1978 The evolution of finite amplitude Kelvin–Helmholtz billows. Geophys. Astrophys. Fluid Dyn. 10, 53–87.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E., 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Riley, J. J. & Metcalfe, R. W., 1980 Direct numerical simulation of a perturbed turbulent mixing layer. AIAA Paper 80-0274.Google Scholar
Stuart, J. T.: 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar
Thorpe, S. A.: 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693–704.Google Scholar
Thorpe, S. A.: 1973 Experiments on stability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731–751.Google Scholar
Thorpe, S. A.: 1985 Laboratory observations of secondary structures in Kelvin–Helmholtz billows and consequences for ocean mixing. Geophys. Astrophys. Fluid Dyn. 34, 175–199.Google Scholar
Thorpe, S. A.: 1987 Transition phenomena and the development of turbulence in stratified fluids. J. Geophys. Res. 92, 5231.Google Scholar
Winant, C. D. & Browand, F. K., 1974 Vortex pairing: the mechanism of turbulent mixing layer growth at moderate Reynolds numbers. J. Fluid Mech. 63, 237–255.Google Scholar
Woods, J. D.: 1969 On Richardson's number as a criterion for laminar–turbulent–laminar transition in the ocean and atmosphere. Radio Sci. 4, 1289–1298.Google Scholar