Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T23:46:30.982Z Has data issue: false hasContentIssue false
  • English
  • Français

On the instability of boundary layers on heated flat plates

Published online by Cambridge University Press:  26 April 2006

Philip Hall  and
Helen Morris
Philip Hall
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
Helen Morris
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a boundary layer on a heated flat plate is investigated in the linear regime. The flow is shown to be unstable to longitudinal vortex structures which develop in a non-parallel manner in the streamwise direction. Solutions of the non-parallel equations are obtained numerically at O(1) values of the appropriate stability parameter, i.e. the Grashof number. We investigate the particular cases in which instability is induced by localized or distributed wall roughness or non-uniform wall heating. The case when the vortices are induced by free-stream disturbances is also considered. We then investigate the high-Grashof-number limit and the fastest growing mode. The fastest growing mode is found to be governed by a quasi-parallel theory and occurs at high wavenumbers. The wavenumber and growth rate of the fastest growing mode are found in closed form. At low wavenumbers the vortex instability is shown to be closely related to Tollmein–Schlichting waves; the effect of wall heating or cooling on the latter type of instability is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 245 , December 1992 , pp. 367 - 400
Copyright
© 1992 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

Akiyama M., Hwang, G. J. & Cheng K. C. 1971 Experiments on the onset of longitudinal vortices in laminar forced convection between horizontal plates. Trans. ASME C: J. Heat Transfer 93, 335–341.Google Scholar
Chen, K. & Cheng M. M. 1984 Thermal instability of forced convection boundary layers. Trans. ASME C: J. Heat Transfer 106, 284–289.Google Scholar
Denier J. P., Hall, P. & Seddougui S. 1991 On the receptivity problem for Görtler vortices: vortex motion induced by wall roughness Phil. Trans. R. Soc. Lond. A 334, 51–85.Google Scholar
Gilpin R. R., Imura, H. & Cheng K. C. 1978 Experiments on the onset of longitudinal vortices in horizontal Blasius flow heated from below. Trans. ASME C: J. Heat Transfer 100, 71–77.Google Scholar
Hall P. 1982a Taylor–Görtler vortices in fully developed or boundary layer flows: linear theory. J. Fluid Mech. 124, 475–494.Google Scholar
Hall P. 1982b On the nonlinear evolution of Görtler vortices in nonparallel boundary layers. J. Inst. Maths Applics. 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. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151–189.Google Scholar
Hall, P. & Smith F. T. 1984 On the effects of nonparallelism, three-dimensionality and mode interaction in boundary layer stability theory. Stud, Appl. Maths 70, 91–120.Google Scholar
Mangalam S. M., Dagenhart, J. R. & Meyers J. F. 1991 Experimental studies on Görtler vortices. NASA Symp. on Natural Laminar Flow and Laminar Flow Control Research. NASA TM (in press).Google Scholar
Moutsoglou A., Chen, T. S. & Cheng K. C. 1981 Vortex Instability of mixed convection flow over a Horizontal Flat Plate. Trans. ASME C: J. Heat Transfer 103, 257–261.Google Scholar
Wang X. A. 1982 An experimental study of mixed, forced and free convection heat transfer from a horizontal plate to air: Trans. ASME C: J. Heat Transfer 104, 139–144.Google Scholar
Wu, R. S. & Cheng K. C. 1976 Thermal instability of Blasius flow along horizontal plates. Intl J. Heat Mass Transfer 105, 907–913.Google Scholar