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The mechanics of the Tollmien-Schlichting wave

Published online by Cambridge University Press:  26 April 2006

Peter G. Baines ,
Sharan J. Majumdar  and
Humio Mitsudera
Peter G. Baines
Affiliation:
CSIRO Division of Atmospheric Research, Aspendale, Australia
Sharan J. Majumdar
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
Humio Mitsudera
Affiliation:
CSIRO Division of Atmospheric Research, Aspendale, Australia Present address: JAMSTEC, Yokosuka, Japan.
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Abstract

We describe a mechanistic picture of the essential dynamical processes in the growing Tollmien-Schlichting wave in a Blasius boundary layer and similar flows. This picture depends on the interaction between two component parts of a disturbance (denoted ‘partial modes’), each of which is a complete linear solution in some idealization of the system. The first component is an inviscid mode propagating on the vorticity gradient of the velocity profile with the free-slip boundary condition, and the second, damped free viscous modes in infinite uniform shear with the no-slip condition. There are two families of these viscous modes, delineated by whether the phase lines of the vorticity at the wall are oriented with or against the shear, and they are manifested as resonances in a forced system. The interaction occurs because an initial ‘inviscid’ disturbance forces a viscous response via the no-slip condition at the wall. This viscous response is large near the resonance associated with the most weakly damped viscous mode, and in the unstable parameter range it has suitable phase at the outer part of the boundary layer to increase the amplitude of the inviscid partial mode by advection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 312 , 10 April 1996 , pp. 107 - 124
Copyright
© 1996 Cambridge University Press

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References

Antar, B. N. & Benek, J. A. 1978 Temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary-layer. Phys. Fluids 21, 183–189.Google Scholar
Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.
Baines, P. G. & Mitsudera, H. 1992 On the mechanism of shear flow instability in stratified fluids and boundary-layers. Proc. 11th Austral. Fluid Mech. Conf., Univ. of Tasmania, Hobart, pp. 75–78.
Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327–342.Google Scholar
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Bogdanova, E. V. & Ryzhov, O. S. 1983 Free and induced oscillations in Poiseuille flow. Q. J. Mech. Appl. Maths 36, 27–87.Google Scholar
Corcos, G. & Sellars, J. R. 1959 On the stability of fully developed flow in a pipe. J. Fluid Mech. 5, 97–112.Google Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Grohne, D. 1954 Über das specktrum bei eigen schwingungen ebener laminarströmungen. Z. Angew Math. Mech. 34, 344–357.Google Scholar
Hartman, R. J. 1975 Wave propagation in a stratified shear flow. J. Fluid Mech. 71, 89–104.Google Scholar
Lighthill, M. J. 1963 Introduction. Boundary layer theory. In Laminar Boundary Layers (ed.L. Rosenhead), pp. 46–113. Oxford University Press.
Lindzen, R. S. 1988 Instability of plane parallel shear flow (towards a mechanistic picture of how it works). Pageoph. 126, 103–121.Google Scholar
Lindzen, R. S. & Rambaldi, S. 1986 A study of viscous Poiseuille flow. J. Fluid Mech. 165, 355–372.Google Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Rayleigh, Lord 1896 The Theory of Sound, vol. ii. Reproduced by Dover (1945).
Sakai, S. 1989 Rossby-Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.Google Scholar
Schlichting, H. 1968 Boundary-Layer Theory. McGraw-Hill.
Smith, F. T. 1979 On the non-parallel flow instability of the Blasius boundary-layer. Proc. R. Soc. Lond A 366, 91–109.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers (ed.L. Rosenhead), pp. 492–579. Oxford University Press.
Takehiro, S. & Hayashi, Y. 1992 Over-reflection and shear instability in a shallow-water model. J. Fluid Mech. 236, 259–279.Google Scholar