Hostname: page-component-669899f699-rg895 Total loading time: 0 Render date: 2025-05-05T12:09:08.121Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave mechanics of breakdown

Published online by Cambridge University Press:  29 March 2006

M. T. Landahl
M. T. Landahl
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

Kinematic wave theory is used to determine under what conditions breakdown of a steady or unsteady laminar flow into high frequency oscillations should occur. The analysis of a small-scale secondary wave riding on a large-scale inhomogeneity, such as that produced by a finite amplitude primary instability wave, reveals that the breakdown mechanism has three basic ingredients: (i) a self-excited secondary wave with a group velocity near the propagation velocity (phase velocity) of the primary wave, (ii) space-time focusing of the secondary wave train on the primary wave crest and (iii) a nonlinear filtering mechanism leading to rectification of the secondary wave.

The theory is applied to a laminar shear flow. Good quantitative agreement with the experiments on boundary-layer transition by Klebanoff, Tidstrom & Sargent (1962) is found for the critical condition leading to breakdown. Also, the theory is able to explain all the main qualitative breakdown features observed by Klebanoff et al. and others, such as the rapid localized onset, and the formation of a hairpin vortex lifting up from the surface downstream of the primary wave crest.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 56 , Issue 4 , 28 December 1972 , pp. 775 - 802
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.)

Article purchase

Temporarily unavailable

References

Barger, W. R., Garrett, W. D., Mollo-Christensen, E. L. & Ruggles, K. W. 1970 J. Appl. Meteor 9, 396.
Benney, D. J. 1961 J. Fluid Mech. 10, 209.
Benney, D. J. & Lin, C. C. 1960 Phys. Fluids, 4, 656.
Betchov, R. 1960 Phys. Fluids, 3, 1026.
Emmons, H. W. 1951 J. Aeron. Sci 18, 490.
Gaster, M. 1968 J. Fluid Mech. 32, 173.
Greenspan, H. & Benney, D. 1963 J. Fluid Mech. 15, 133.
Hama, F. R. & Nutant, J. 1963 Proc. 1963 Heat Transfer and Fluid Mech. Inst. p. 77. Stanford University Press.
Hayes, W. D. 1970 Proc. Roy. Soc. A 320, 209.
Kim, H. T., Kline, S. J. & Reynolds, W. C. 1971 J. Fluid Mech. 50, 133.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. H. 1962 J. Fluid Mech. 12, 1.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 J. Fluid Mech. 30, 741.
Kovasznay, L. S. G., Komoda, H. S. & Vastjdeva, B. R. 1962 Proc. 1962 Heat Transfer and Fluid Mech. Inst., p. 1. Stanford University Press.
Landahl, M. T. 1967 J. Fluid Mech. 29, 441.
Landahl, M. T. 1969 Phys. Fluids, 12 (suppl. II), 146.
Lighthill, M. J. 1969 In Computation of Turbulent Boundary Layers (ed. S. J. Kline, M. V. Morkovin, G. Sovram & D. J. Cookrell), vol. 1, p. 511. Stanford University.
Lighthill, M. J. 1970 Osborne Reynolds and Engineering Science, p. 83. Manchester University Press.
Miles, J. W. 1959 J. Fluid Mech. 6, 568.
Mollo-Christensen, E. L. 1971 A.I.A.A. J. 7, 1217.
Obremski, H. J. & Fejer, A. A. 1967 J. Fluid Mech. 29, 93.
Obremski, H. J. & Morkovin, M. V. 1969 A.I.A.A. J. 7, 1298.
Obremski, H. J., Morkovin, M. V. & Landahl, M. T. 1969 AGARDOgraph, no. 134.
Reynolds, O. 1883 Phil. Trans. Roy. Soc. 174, 935.
Schauber, G. B. & Skramstad, H. K. 1948 Nat. Adv. Comm. Aeron. Rep. no. 909.
Whitham, G. B. 1965 Proc. Roy. Soc. A 283, 238.