Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-08T09:29:56.950Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly nonlinear wave motions in a thermally stratified boundary layer

Published online by Cambridge University Press:  26 April 2006

James P. Denier  and
Eunice W. Mureithi
James P. Denier
Affiliation:
Department of Applied Mathematics, University of Adelaide, South Australia 5005, Australia
Eunice W. Mureithi
Affiliation:
Department of Applied Mathematics, University of Adelaide, South Australia 5005, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider weakly nonlinear wave motions in a thermally stratified boundary layer. Attention is focused on the upper branch of the neutral stability curve, corresponding to small wavelengths and large Reynolds number. In this limit the motion is governed by a first harmonic/mean flow interaction theory in which the wave-induced mean flow is of the same order of magnitude as the wave component of the flow. We show that the flow is governed by a system of three coupled partial differential equations which admit finite-amplitude periodic solutions bifurcating from the linear, neutral points.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 315 , 25 May 1996 , pp. 293 - 316
Copyright
© 1996 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. 1965 Handbook of Mathematical Functions. Dover.
Bassom. A. P. & Hall, P. 1989 On the generation of mean flows by the interaction of Görtler vortices and Tollmien-Schlichting waves in curved channel flows. Stud. Appl. Maths 81, 185.Google Scholar
Benney, D. J. & Chow, K. 1985 An alternative approach to nonlinear instabilities in hydrodynamics. Stud. Appl. Maths 73, 261–267.Google Scholar
Benney, D. J. & Chow, K 1986 Instabilities of waves on a fee surface. Stud. Appl. Maths 74, 227–243.Google Scholar
Benney, D. J. & Chow, K. 1989 A mean flow first harmonic theory for hydrodynamic stability. Stud. Appl. Maths 80, 37–73.Google Scholar
Bodonyi, R. J. & Smith, F. T. 1981 The upper branch stability of the Blasius boundary layer, including non-parallel flow effects. Proc. R. Soc. Lond. A 375, 65–92.Google Scholar
Denier, J. P. 1992 The development of fully nonlinear Taylor vortices. IMA J. Appl. Maths 49, 15–33.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Gage, K. S. 1971 The effect of stable thermal stratification on the stability of viscous parallel flows. J. Fluid Mech. 47, 1–20.Google Scholar
Gage, K. S. & Reid, W. H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33, 21–32.Google Scholar
Hall, P. 1982 Taylor-Görtler vortices in fully developed or boundary layer flows: linear theory. J. Fluid Mech. 124, 475–494.Google Scholar
Hall, P. 1983 On the non-linear evolution of Görtler vortices in non-parallel boundary layers. IMA J. Appl. Maths 29, 173–196.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. 1993 Streamwise vortices in heated boundary layers. J. Fluid Mech. 252, 301–324.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. & Morris, H. 1992 On the instability of boundary layers on the heated flat plates. J. Fluid Mech. 245, 367–400.Google Scholar
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Mureithi, E. W. 1996 The effect of thermal stratification on the stability properties of boundary layer flows. PhD thesis, University of New South Wales.
Mureithi, E. W. & Denier, J. P. 1996 The effect of buoyancy on lower branch Tollmien-Schlichting waves: weakly nonlinear theory (in preparation).
Mureithi, E. W., Denier, J. P. & Stott, J. A. K. 1996 The effect of buoyancy on upper branch Tollmien-Schlichting waves. IMA J. Appl. Maths (submitted).Google Scholar
Reid, W.H. 1965 The stability of parallel flows. In Basic Developments in Fluid Dynamics, vol. 1 (ed. M. Holt), pp. 249–307. Academic.
Seddougui, S. O., Bowles, R. I. & Smith, F. T. 1981 Surface-cooling effects on compressible boundary-layer instability, and on upstream influence. Eur. J. Mech. B/Fluids 10, 117–145.Google Scholar
Smith, F. T. 1979a On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T. 1979b Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573–589.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1980 On the stability of the developing flow in a channel or circular pipe. Q. J. Mech. Appl. Maths 33, 293–320.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982 Nonlinear critical layers and their development in streaming-flow stability. J. Fluid Mech. 118, 165–185.Google Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Clarendon.
Strehle, E. 1978 Stabilitätsberechnung dichtegeschichteter ebener Wandgrenzschichten. Z. Angew. Math. Mech. 58, 539–552.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers (ed. L. Rosenhead). Clarendon Press.