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Another route to the three-dimensional development of Tollmien-Schlichting waves with finite amplitude

Published online by Cambridge University Press:  21 April 2006

Nobutake Itoh
Nobutake Itoh
Affiliation:
National Aerospace Laboratory, Chofu, Tokyo, Japan
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Abstract

The Tollmien-Schlichting waves appearing as a result of instability of laminar flows develop a three-dimensional configuration as the amplitude becomes large enough. A new explanation of this experimentally observed phenomenon is attempted on the basis of a resonance theory. It is shown that the existence of two-dimensional waves with finite amplitude can induce three-dimensional distortion with spanwise periodicity of the mean-flow field. Under a certain condition for resonance, the distortion grows, in proportion to the product of time and an exponential function of time, up to quite a large magnitude, and consequently interacts with the Tollmien-Schlichting waves to yield new three-dimensional travelling waves with the same streamwise wavenumber as the two-dimensional waves, and with the same spanwise wavenumber as the mean flow. The resulting flow field is of the peakvalley-splitting type, as observed often in experiments. The growth rate of the three-dimensional part in the mean flow depends significantly upon values of the spanwise wavenumber, suggesting that there is a preferred range of spanwise periodicity in the three-dimensional development of unstable laminar flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 181 , August 1987 , pp. 1 - 16
Copyright
© 1987 Cambridge University Press

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References

Benney, D. J. 1984 Stud. Appl. Maths 70, 1.
Benney, D. J. & Gustavsson, L. H. 1981 Stud. Appl. Maths 64, 185.
Benney, D. J. & Lin, C. C. 1960 Phys. Fluids 3, 656.
Craik, A. D. D. 1971 J. Fluid Mech. 50, 393.
Craik, A. D. D. 1982 J. Fluid Mech. 125, 37.
Davey, A. & Nguyen, H. P. F. 1971 J. Fluid Mech. 45, 701.
Dhanak, M. R. 1983 Proc. R. Soc. Lond. A 385, 53.
Herbert, Th. 1980 AIAA J. 18, 243.
Herbert, Th. 1983 J. Fluid Mech. 126, 167.
Herbert, Th. 1984a In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), p. 53. North-Holland.
Herbert, Th. 1984b AGARD Rep. 709, p. 7–1.
Herbert, Th. 1985 In Laminar-Turbulent Transition (ed. V. V. Kozlov), p. 9. Springer.
Herbert, Th. & Morkovin, M. V. 1980 In Laminar-Turbulent Transition (ed. R. Eppler & H. Fasel), p. 47. Springer.
Itoh, N. 1974a Trans. Japan Soc. Aero. Space Sci. 17, 65.
Itoh, N. 1974b Trans. Japan Soc. Aero. Space Sci. 17, 160.
Itoh, N. 1974c Trans. Japan Soc. Aero. Space Sci. 17, 175.
Itoh, N. 1977 J. Fluid Mech. 82, 455.
Itoh, N. 1980 Trans. Japan Soc. Aero. Space Sci. 23, 91.
Itoh, N. 1984 In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), p. 59. North-Holland.
Kachanov, Y. S. & Levchenko, V. Y. 1984 J. Fluid Mech. 138, 209.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 J. Fluid Mech. 12, 1.
Mizushima, J. & Gotoh, K. 1985 J. Phys. Soc. Japan 54, 2061.
Nayfeh, A. H. 1981 J. Fluid Mech. 107, 441.
Nishioka, M. & Asai, M. 1985 In Laminar-Turbulent Transition (ed. V. V. Kozlov), p. 173. Springer.
Raetz, G. S. 1959 Norair Rep. NOR-59–383, Hawthorne, California.
Reynolds, W. C. & Potter, M. C. 1967 J. Fluid Mech. 27, 465.
Saric, W. S. & Thomas, A. S. W. 1984 In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), p. 117. North-Holland.
Stuart, J. T. 1960 J. Fluid Mech. 9, 353.
Stuart, J. T. 1962 Adv. Aeronaut. Sci. 3, 121.
Tani, I. 1969 Ann. Rev. Fluid Mech. 1, 169.
Watson, J. 1960 J. Fluid Mech. 9, 371.