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Taylor-Görtler instabilities of Tollmien-Schlichting waves and other flows governed by the interactive boundary-layer equations

Published online by Cambridge University Press:  21 April 2006

Philip Hall  and
James Bennett
Philip Hall
Affiliation:
Mathematics Department, North Park Road, University of Exeter, Exeter, UK
James Bennett
Affiliation:
Mathematics Department, North Park Road, University of Exeter, Exeter, UK
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Abstract

The Taylor-Görtler vortex instability equations are formulated for steady and unsteady interacting boundary-layer flows. The effective Görtler number is shown to be a function of the wall shape in the boundary layer and the possibility of both steady and unsteady Taylor-Görtler modes exists. As an example the steady flow in a symmetrically constricted channel is considered and it is shown that unstable Görtler vortices exist before the boundary layers at the wall develop the Goldstein singularity discussed by Smith & Daniels (1981). As an example of an unsteady spatially varying basic state we also consider the instability of high-frequency large-amplitude two- and three-dimensional Tollmien-Schlichting waves in a curved channel. It is shown that they are unstable in the first ‘Stokes-layer stage’ of the hierarchy of nonlinear states discussed by Smith & Burggraf (1985). This instability of Tollmien-Schlichting waves in an internal flow can occur in the presence of either convex or concave curvature. Some discussion of this instability in external flows is given.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 171 , October 1986 , pp. 441 - 457
Copyright
© 1986 Cambridge University Press

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References

Davey A.1962 The growth of Taylor vortices in flow between rotating cylinders. J. Fluid Mech. 14, 336.Google Scholar
Dean W. R.1928 Fluid motion in a curved channel Proc. R. Soc. Lond. A 121, 402.Google Scholar
Floryan, J. & Saric W. S.1979 Stability of Görtler vortices in boundary layers. AIAA Paper 79–1497.Google Scholar
Görtler H.1940 On the three-dimensional instability of laminar boundary layers on concave walls. Tech. Memor. Natl Adv. Comm. Aero. No. 1375.Google Scholar
Hall P.1982 Taylor-Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475.Google Scholar
Hall P.1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41.Google Scholar
Hall P.1984 On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347.Google Scholar
Hall P.1985 The Görtler vortex instability mechanism in three-dimensional boundary layers Proc. R. Soc. Lond. A 399, 135.Google Scholar
Papageorgiou D.1986 Instability of unsteady flows in curved pipes. J. Fluid Mech. (submitted).Google Scholar
Park K., Barenghi, C. & Donnelly R. J.1980 Subharmonic destabilization of Taylor-Görtler vortices near an oscillating cylinder. Phys. Lett. 78a, 152.Google Scholar
Schlichting H.1932 Berechnung ebener periodischer Grenzschichtströmungen. Z. Phys. 33, 327.Google Scholar
Seminara, G. & Hall P.1976 Centrifugal instability of a Stokes layer: linear theory Proc. R. Soc. Lond. A 350, 299.Google Scholar
Smith A. M. D.1955 On the growth of Taylor-Görtler vortices along highly concave walls. Q. Appl. Maths 13, 233.Google Scholar
Smith F. T.1979 Instability of flow through pipes of general cross section. Mathematica 26, 187.Google Scholar
Smith F. T.1986 The strong non-linear growth of three-dimensional disturbances in boundary layers. J. Fluid Mech. (submitted).Google Scholar
Smith, F. T. & Bodonyi R.1985 On short-scale inviscid instabilities in flow past surface-mounted obstacles and other non-parallel motions. Aero. Quart. 82, 205.Google Scholar
Smith, F. T. & Burggraf O.1985 On the development of large-sized short-scaled disturbances in boundary layers Proc. R. Soc. Lond. A 399, 25.Google Scholar
Smith, F. T. & Daniels P. G.1981 Removal of Goldstein's singularity at separation, in flow part obstacles in wall layers. J. Fluid Mech. 110, 1.Google Scholar
Stephanoff K. D., Pedley T. J., Lawrence, C. J. & Secomb T. W.1983 Fluid flow along a channel with an asymmetric oscillating constriction. Nature 305, 692.Google Scholar
Tutty, O. & Cowley S.1986 On the stability and the numerical solution of the unsteady interactive boundary-layers equation. J. Fluid Mech. 168, 431.Google Scholar