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A phase-equation approach to boundary–layer instability theory: Tollmien-Schlichting waves

Published online by Cambridge University Press:  26 April 2006

Philip Hall
Philip Hall
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
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Abstract

Our concern is with the evolution of large-amplitude Tollmien-Schlichting waves in boundary-layer flows. In fact, the disturbances we consider are of a comparable size to the unperturbed state. We shall describe two-dimensional disturbances which are locally periodic in time and space. This is achieved using a phase equation approach of the type discussed by Howard & Kopell (1977) in the context of reaction-diffusion equations. We shall consider both large and O(1) Reynolds number flows though, in order to keep our asymptotics respectable, our finite-Reynolds-number calculation will be carried out for the asymptotic suction flow. Our large-Reynolds-number analysis, though carried out for Blasius flow, is valid for any steady two-dimensional boundary layer. In both cases the phase-equation approach shows that the wavenumber and frequency will develop shocks or other discontinuities as the disturbance evolves. As a special case we consider the evolution of constant frequency/wavenumber disturbances and show that their modulational instability is controlled by Burgers equation at finite-Reynolds-number and by a new integro-differential evolution equation at large-Reynolds-numbers. For the large Reynolds number case the evolution equation points to the development of a spatially localized singularity at a finite time.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 304 , 10 December 1995 , pp. 185 - 212
Copyright
© 1995 Cambridge University Press

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References

Brotherton-Ratcliffe, R. V. & Smith, F. T. 1987 Mathematika 34, 86.
Conlisk, A. T., Burcxjraf, O. R., Smith, F. T. 1987 Trans. ASME FED, vol. 52, p. 119.
Cross, M. & Newell, A. C. 1984 Physica D 10, 299.
Davey, A., Hocking, L. M. & Stewartson, K. 1974 J. Fluid Mech. 63, 529.
DiPRiMA, R. C. & Stuart, J. T. 1975 J. Fluid Mech. 67, 85.
Gaster, M. 1974 J. Fluid Mech. 66, 465.
Hall, P. 1988 J. Fluid Mech. 193, 243.
Hall, P. & Smith, F. T. 1984 Stud. Appl. Maths 70, 91.
Herbert, T. 1977 AGARD Conf. Proc. CP 224, 3/1-10.Google Scholar
Howard, L. & Kopell, N. 1977 Stud. Appl. Maths 56, 95.
Hocking, L. M. 1975 Q. J. Mech. Appl. Maths 28, 341.
Kachanov, Yu. S. & Levchenko, V. Yu 1984 J. Fluid Mech. 138, 209.
Khokhlov, A. P. 1994 J. Appl. Maths Tech. Phys. 34, 508.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 J. Fluid Mech. 12, 1.
Kramer, L., Ben Jacob, E., Brand, H. & Coss, M. C. 1982 Phys. Rev. Lett. 49, 1891.
Lin, C. C. 1966 The Theory of Hydrodynamic Stability. Cambridge University Press.
Newell, A. C., Passot, T. & Lega, J. 1993 Ann. Rev. Fluid Mech. 25, 399.
Papageorgiu, D. T. & Smyrlis, Y. S. 1991 Proc. Natl Acad. Sci. USA, 24, 11129.
Smith, F. T. 1973 J. Fluid Mech. 57, 803.
Smith, F. T. 1979a Proc. R. Soc. Lond. A 366, 91.
Smith, F. T. 1979 Proc. R. Soc. Lond. A 368, 513.
Smith, F. T. & Burggraf, O. R. 1985 Proc. R. Soc. Lond. A 399, 25.
Smith, F. T. & Stewart, P. 1987 J. Fluid Mech. 194, 21.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Wray, A. A. & Hussaini, M. Y. 1984 Proc. R. Soc. Lond. A 392, 373.
Zhuk, V. I. & Ryzhov, O. S. 1982 Sov. Phys. Dokl. 27, 177.