Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-18T20:52:14.393Z Has data issue: false hasContentIssue false
  • English
  • Français

The two- and three-dimensional instabilities of a spatially periodic shear layer

Published online by Cambridge University Press:  20 April 2006

R. T. Pierrehumbert  and
S. E. Widnall
R. T. Pierrehumbert
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 Present address: Department of Meteorology, Massachusetts Institute of Technology, Cambridge, MA 02139.
S. E. Widnall
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

The two- and three-dimensional stability properties of the family of coherent shear-layer vortices discovered by Stuart are investigated. The stability problem is formulated as a non-separable eigenvalue problem in two independent variables, and solved numerically using spectral methods. It is found that there are two main classes of instabilities. The first class is subharmonic, and corresponds to pairing or localized pairing of vortex tubes; the pairing instability is most unstable in the two-dimensional limit, in which the perturbation has no spanwise variations. The second class repeats in the streamwise direction with the same periodicity as the basic flow. This mode is most unstable for spanwise wavelengths approximately 2/3 of the space between vortex centres, and can lead to the generation of streamwise vorticity and coherent ridges of upwelling. Comparison is made between the calculated instabilities and the observed pairing, helical pairing, and streak transitions. The theoretical and experimental results are found to be in reasonable agreement.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 114 , January 1982 , pp. 59 - 82
Copyright
© 1972 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

Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in shear flow. Phys. Fluids 3, 656.Google Scholar
Bernal, L. P., Breidenthal, R. E., Brown, G. L., Konrad, J. H. & Roshko, A. 1979 On the development of three dimensional small scales in turbulent mixing layers. In Proc. 2nd Int. Symp. on Turbulent Shear Flows, Imperial College, London.
Bliss, D. B. 1973 Ph.D. dissertation, Massachusetts Institute of Technology.
Boyd, J. 1978 Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems. Monthly Weather Rev. 106, 1192–1203.Google Scholar
Breidenthal, R. E. 1978 A chemically reacting turbulent shear layer. Ph.D. thesis, California Institute of Technology.
Browand, F. K. & Troutt, T. 1980 A note on spanwise structure in the two-dimensional mixing layer. J. Fluid Mech. 97, 771.Google Scholar
Browand, F. K. & Weidman, P. D. 1976 Large scales in the developing mixing layer. J. Fluid Mech. 76, 127.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structures in turbulent mixing layers. J. Fluid Mech. 64, 775.Google Scholar
Chandrsuda, C., Mehta, R. D., Weir, A. D. & Bradshaw, P. 1978 Effect of free-stream turbulence on large structure in turbulent mixing layers. J. Fluid Mech. 85, 693.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications, S.I.A.M. Regional Conf. Series in Appl. Math., vol. 26.
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn, pp. 156, 224. Dover.
Mansour, N. N., Ferziger, J. & Reynolds, W. C. 1978 Large eddy simulation of a turbulent mixing layer. Thermosci. Rep. TF-11 Div., Dept. of Mech. Engng, Stanford Univ.
Miksad, R. W. 1972 Experiments on the nonlinear stages of free shear layer transition. J. Fluid Mech. 56, 645.Google Scholar
Moore, D. W. & Saffman, P. G. 1975a The density of organized vortices in a turbulent mixing layer. J. Fluid Mech. 69, 465.Google Scholar
Moore, D. W. & Saffman, P. G. 1975b The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413.Google Scholar
Orszag, S. A. & Patera, A. 1980 Subcritical transitions to turbulence in plane channel flows. Phys. Rev. Lett. 45, 989.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1981 The structure of organized vortices in a free shear layer. J. Fluid Mech. 102, 301.Google Scholar
Roshko, A. 1976 Structure of Turbulent Shear Flows: A New Look. A.I.A.A. J. 14, 1349.
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
Tsai, C. Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixinglayer growth at moderate Reynolds number. J. Fluid Mech. 63, 237.Google Scholar
Wygnanski, I., Oster, D., Fiedler, H. & Dziomba, B. 1979 On the perseverance of a quasi-two-dimensional eddy-structure in a turbulent mixing layer. J. Fluid Mech. 93, 325.Google Scholar