Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-19T03:20:42.489Z Has data issue: false hasContentIssue false
  • English
  • Français

The onset of turbulence in finite-amplitude Kelvin–Helmholtz billows

Published online by Cambridge University Press:  20 April 2006

G. P. Klaassen  and
W. R. Peltier
G. P. Klaassen
Affiliation:
Department of Physics, University of Toronto, Ontario, Canada M5S1A7
W. R. Peltier
Affiliation:
Department of Physics, University of Toronto, Ontario, Canada M5S1A7
Get access Rights & Permissions [Opens in a new window]

Abstract

Two-dimensional finite-amplitude Kelvin–Helmholtz waves are tested for stability against three-dimensional infinitesimal perturbations. Since the nonlinear waves are time-dependent, the stability analysis is based upon the assumption that they evolve on a timescale which is long compared with that of any instability which they might support. The stability problem is thereby reduced to standard eigenvalue form, and solutions that do not satisfy the timescale constraint are rejected. If the Reynolds number of the initial parallel flow is sufficiently high the two-dimensional wave is found to be unstable and the fastest-growing modes are three-dimensional disturbances that possess longitudinal symmetry. These modes are convective in nature and focused in the statically unstable regions that form during the overturning of the stratified fluid in the core of the nonlinear vortex. The nature of the instability in the high-Reynolds-number regime suggests that it is intimately related to the observed onset of turbulence in these waves. The transition Reynolds number above which the secondary instability exists depends strongly on the initial conditions from which the primary wave evolves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 155 , June 1985 , pp. 1 - 35
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

Boyd, J. 1978 Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems. Mon. Weather Rev. 106, 1192.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775.Google Scholar
Busse, F. H. 1972 The oscillatory instability of convection rolls in a low Prandtl number fluid. J. Fluid Mech. 52, 97.Google Scholar
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625.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, 2394.2.0.CO;2>CrossRefGoogle Scholar
Denny, V. E. & Clever, R. M. 1974 Comparison of Galerkin and finite difference methods for solving highly nonlinear thermally driven flows. J. Comp. Phys. 16, 271.Google Scholar
Jordan, D. W. & Smith, P. 1977 Nonlinear Ordinary Differential Equations. Oxford University Press.
Keller, H. B. 1976 Numerical Solution of Two Point Boundary Value Problems. SIAM Regional Conf. Ser. in Appl. Maths, vol. 24. Arrowsmith.
Kelly, R. E. 1977 The onset and development of Rayleigh–Bénard convection in shear flows: a review. In Physiochemical Hydrodynamics, vol. 1 (ed. V. G. Levich). Advance.
Klaassen, G. P. 1982 The transition to turbulence in stably stratified parallel flows. Ph.D. Thesis, University of Toronto.
Klaassen, G. P. & Peltier, W. R. 1985a The evolution of finite amplitude Kelvin–Helmholtz billows in two spatial dimensions. J. Atmos. Sci. (in press).Google Scholar
Klaassen, G. P. & Peltier, W. R. 1985b The effect of the Prandtl number on the evolution and stability of finite amplitude Kelvin–Helmholtz billows. Geophys. Astrophys. Fluid Dyn. (in press).Google Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135.Google Scholar
Ogura, Y. & Phillips, N. A. 1962 Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173.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.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.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167 (and Addendum 23, 343).Google Scholar
Scorer, R. S. 1969 Mechanisms of clear air turbulence. In Clear Air Turbulence and its Detection (ed. Y. H. Pao & A. Goldburg). Plenum.
Smith, B. T., Boyle, J. M., Dongarron, J. J., Garbow, B. S., Ikebe, Y., Klema, V. C. & Moler, C. B. 1974 Matrix Eigensystem Routines – EISPACK Guide. Lecture Notes in Computer Science, vol. 6. Springer.
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417.Google Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731.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.Google Scholar