Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-17T02:23:00.641Z Has data issue: false hasContentIssue false
  • English
  • Français

The influence of stratification on secondary instability in 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

We analyse the stability of horizontally periodic, two-dimensional, finite-amplitude Kelvin-Helmholtz billows with respect to infinitesimal three-dimensional perturbations having the same streamwise wavelength for several different levels of the initial density stratification. A complete analysis of the energy budget for this class of secondary instabilities establishes that the contribution to their growth from shear conversion of the basic-state kinetic energy is relatively insensitive to the strength of the stratification over the range of values considered, suggesting that dynamical shear instability constitutes the basic underlying mechanism. Indeed, during the initial stages of their growth, secondary instabilities derive their energy predominantly from shear conversion. However, for initial Richardson numbers between 0.065 and 0.13, the primary source of kinetic energy for secondary instabilities at the time the parent wave climaxes is in fact the conversion of potential energy by convective overturning in the cores of the individual billows. A comparison between the secondary instability properties of unstratified Kelvin-Helmholtz billows and Stuart vortices is made, as the latter have often been assumed to provide an adequate approximation to the former. Our analyses suggest that the Stuart vortex model has several shortcomings in this regard.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 71 - 106
Copyright
© 1991 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

Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359–391.Google Scholar
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
Brown, G. L. & Roshko, A. 1974 On density effects and large scale structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.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
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
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
Klaassen, G. P. 1991 Subharmonic secondary instabilities in free shear layers. Geophys. Astrophys. Fluid Dyn. (sub judice).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. Feb. 3–5, 1987, Pasadena, CA, vol. I. (Reprinted in Stratified Flows (ed. E. J. List & G. H. Jirka). ASME.)
Klaassen, G. P. & Peltier, W. R. 1989 The role of transverse secondary instabilities in the evolution of free shear layers. J. Fluid Mech. 202, 367–402.Google Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135–159.Google Scholar
Laprise, R. & Peltier, W. R. 1989 The linear stability of nonlinear mountain waves: Implications for the understanding of severe down slope wind storms. J. Atmos. Sci. 46, 545–564.Google Scholar
Lasheras, J. C., Cho, J. S. & Maxworthy, T. 1986 On the origin and evolution of streamwise vortical structures in a plane free shear-layer J. Fluid Mech. 172, 231–258.Google Scholar
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
Lin, S. J. & Corcos, G. M. 1984 The mixing layer: deterministic models of a turbulent flow. Part 3. The effect of plane strain on the dynamics of streamwise vortices. J. Fluid Mech. 141, 139–178.Google Scholar
Maslowe, S. A. 1973 Finite amplitude Kelvin-Helmholtz billows. Boundary-Layer Met. 5, 43–52.Google Scholar
Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S. & Riley, J. J. 1987 Secondary instability of a temporally growing mixing layer. J. Fluid Mech. 184, 207–243.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. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.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 (referred to herein as PW82).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. 1971 Experiments on the instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299–320.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