Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T23:02:48.940Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex instabilities in three-dimensional boundary layers: the relationship between Görtler and crossflow vortices

Published online by Cambridge University Press:  26 April 2006

Andrew P. Bassom  and
Philip Hall
Andrew P. Bassom
Affiliation:
Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE, UK
Philip Hall
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The inviscid and viscous stability problems are addressed for a boundary layer which can support both Görtler and crossflow vortices. The change in structure of Görtler vortices is found when the parameter representing the degree of three-dimensionality of the basic boundary-layer flow under consideration is increased. It is shown that crossflow vortices emerge naturally as this parameter is increased and ultimately become the only possible vortex instability of the flow. It is shown conclusively that at sufficiently large values of the crossflow there are no unstable Görtler vortices present in a boundary layer which, in the zero-crossflow case, is centrifugally unstable. The results suggest that in many practical applications Görtler vortices cannot be a cause of transition because they are destroyed by the three-dimensional nature of the basic state. In swept-wing flows the Görtler mechanism is probably not present for typical angles of sweep of about 20°.

Some discussion of the receptivity problem for vortex instabilities in weakly three-dimensional boundary layers is given; it is shown that inviscid modes have a coupling coefficient marginally smaller than those of the fastest growing viscous modes discussed recently by Denier, Hall & Seddougui (1991). However, the fact that the growth rates of the inviscid modes are the larger in most situations means that they are probably the more likely source of transition.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 647 - 680
Copyright
© 1991 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.)

References

Abramowitz, M. & Stegun, I. A. 1964 A Handbook of Mathematical Functions. Frankfurt: National Bureau of Standards.
Bassom, A. P. 1989 On the effect of crossflow on nonlinear Götler vortices in curved channel flows. Q. J. Mech. Appl. Maths 42, 495–510.Google Scholar
Bassom, A. P. & Gajjabs, J. S. B. 1988 Non-stationary cross-flow vortices in three-dimensional boundary layer flows. Proc. R. Soc. Land. A 417, 179–212.Google Scholar
Bassom, A. P. & Seddougui, S. O. 1990 The onset of three-dimensionality and time-dependence in Görtler vortices: neutrally stable wavy modes. J. Fluid Mech. 220, 661–672.Google Scholar
Cookes, J. C. 1950 The boundary layer of a class of infinite yawed cylinders. Proc. Camb. Phil. Soc. 46, 645–648.Google Scholar
Denier, J. P., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 335, 51–85 (referred to herein as DHS).Google Scholar
Görtler, H. 1940 Uber eine dreidimensionale instabilitat laminare Grenzschubten an Konkaven Wänden. NACA TM 1357.
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155–199.Google Scholar
Hall, P. 1982a Taylor—Görtler vortices in fully developed or boundary layer flows. J. Fluid Mech. 124, 475–494.Google Scholar
Hall, P. 1982b On the nonlinear evolution of Görtler vortices in non-parallel boundary layers. J. Inst. Maths Applies 29, 173–196.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Hall, P. 1985 The Görtler vortex instability mechanism in three-dimensional boundary layers. Proc. R. Soc. Lond. A 399, 135–152.Google Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 247–266.Google Scholar
Hall, P. & Lakin, W. D. 1988 The fully nonlinear development of Görtler vortices in growing boundary layers. Proc. R. Soc. Lond. A 415, 421–444.Google Scholar
Hall, P. & Seddougui, S. 1989 On the onset of three-dimensionality and time-dependence in Görtler vortices. J. Fluid Mech. 204, 405–420.Google Scholar
Hämmerlin, G. 1956 Zur Theorie der dreidimensionalen Instabilitat laminar Grenschichten. Z. Angew. Math. Phys. 1, 156–167.Google Scholar
Jones, C. W. & Watson, E. J. 1963 Two-dimensional boundary layers. In Laminar Boundary Layers (ed. L. Rosenhead), Chap. 5. Oxford University Press.
Malik, M. R., Chuang, S. & Hussaini, M. Y. 1982 Accurate numerical solution of compressible, linear stability equations. Z. Angew. Math. Phys. 33, 189–201.Google Scholar
Peerhossaini, H. & Wesfreid, J. E. 1988a On the inner structure of streamwise Görtler rolls. Intl J. Heat Fluid Flow 9, 12–18.Google Scholar
Peebhossaini, H. & Wesfreid, J. E. 1988b Experimental study of the Taylor—Görtler instability. In Propagation in Systems Far from Equilibrium (ed. J. E. Wesfreid, H. B. Brand, P. Manneville, G. Albinet & N. Boccara). Springer Series in Synergetics, vol. 41, pp. 399–412. Springer.
Seddotjgui, S. O. & Bassom, A. P. 1991 On the instability of Görtler vortices to non-linear travelling waves. IMA J. Appl. Maths 46, 269–296.Google Scholar
Smith, A. M. O. 1955 On the growth of Taylor—Görtler vortices along highly concave walls. Q. Appl. Maths 13, 233–262.Google Scholar