Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-12T19:34:35.355Z Has data issue: false hasContentIssue false
  • English
  • Français

Neutrally stable wave motions in thermally stratified Poiseuille-Couette flow

Published online by Cambridge University Press:  17 February 2009

James P. Denier  and
Andrew P. Bassom
James P. Denier
Affiliation:
Department of Applied Mathematics, University of Adelaide, South Australia, 5005, Australia, email:jdenier@maths.adelaide.edu.au
Andrew P. Bassom
Affiliation:
2Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE, United Kingdom, email:drew@maths.exeter.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The influence of thermal buoyancy on neutral wave modes in Poiseuille-Couette flow is considered. We examine the modifications to the asymptotic structure first described by Mureithi, Denier & Stott [16], who demonstrated that neutral wave modes in a strongly thermally stratified boundary layer are localized at the position where the streamwise velocity attains its maximum value. The present work demonstrates that such a flow structure also holds for Poiseuille-Couette flow but that a new flow structure emerges as the position of maximum velocity approaches the wall (and which occurs as the level of shear, present as a consequence of the Couette component of the flow, is increased). The limiting behaviour of these wave modes is discussed thereby allowing us to identify the parameter regime appropriate to the eventual restabilization of the flow at moderate levels of shear.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 1 , July 1998 , pp. 123 - 144
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Abramowitz, M. and Stegun, I. A. (eds.). Handbook of Mathematical Functions (Dover, New York, 1965).Google Scholar
[2]Blackaby, N. D. and Choudhari, M., “Inviscid vortex motions in weakly three-dimensional boundary layers and their relation with instabilities in stratified shear flows”, Proc. R. Soc. Lond. A 440 (1993) 701–710.Google Scholar
[3]Cowley, S. J. and Smith, F. T., “On the stability of Poiseuille–Couette flow: a bifurcation from infinity”, J. Fluid Mech. 156 (1985) 83–100.CrossRefGoogle Scholar
[4]Deardorff, J. W., “Gravitational instability between horizontal plates with shear”, Phys. Fluids 8 (1965) 1027–1030.CrossRefGoogle Scholar
[5]Denier, J. P., “Nonlinear wave interactions in stratified Poiseuille–Couette flow” (1998), in preparation.Google Scholar
[6]Denier, J. P. and Mureithi, E. W., “Weakly nonlinear wave motions in a thermally stratified boundary layer”, J. Fluid Mech. 315 (1996) 293–316.CrossRefGoogle Scholar
[7]Drazin, P. G. and Reid, W. H., Hydrodynamic stability (C.U.P., 1979).Google Scholar
[8]Fujimura, K. and Kelly, R. E., “Stability of unstably stratified shear flow between parallel plates”, Fluid Dyn. Res. 2 (1988) 281–292.CrossRefGoogle Scholar
[9]Gage, K. S., “The effect of stable thermal stratification on the stability of viscous parallel flows”, J. Fluid Mech. 47 (1974) 1–20.CrossRefGoogle Scholar
[10]Gage, K. S. and Reid, W. H., “The stability of thermally stratified plane Poiseuille flow”, J. Fluid Mech. 33 (1968) 21–32.Google Scholar
[11]Gallagher, A. P. and Mercer, A. McD., “On the behaviour of small disturbances in plane Couette flow with a temperature gradient”, Proc. Roy. Soc. Lond. A 286 (1965) 117–128.Google Scholar
[12]Hall, P., “Taylor-Gortler vortices in fully developed or boundary layer flows: linear theory”, J. Fluid Mech. 124 (1982) 475–494.CrossRefGoogle Scholar
[13]Hughes, T. H. and Reid, W. H., “The stability of spiral flow between rotating cylinders”, Phil. Trans. R. Soc. Lond. 263 (1968) 57–91.Google Scholar
[14]Koppel, D., “On the stability of flow of a thermally stratified fluid under the action of gravity”, J. Math. Phys. 5 (1964) 963–982.CrossRefGoogle Scholar
[15]Miles, J. W., “On the stability of heterogeneous shear flows”, J. Fluid Mech. 10 (1961) 496–508.CrossRefGoogle Scholar
[16]Mureithi, E. W., Denier, J. P. and Stott, J. A. K., “The effect of buoyancy on upper branch Tollmien–Schlichting waves”, IMA. J. Appl. Math. 58 (1997) 19–50.CrossRefGoogle Scholar
[17]Otto, S. R. and Bassom, A. P., “Weakly nonlinear stability of viscous vortices in three-dimensional boundary layers”, J. Fluid Mech. 249 (1993) 597–618.Google Scholar
[18]Schäfer, P. and Herwig, H., “Stability of plane Poiseuille flow with temperature dependent viscosity”, Int. J. Heat Mass Transfer 36 (1993) 2441–2448.CrossRefGoogle Scholar
[19]Tveitereid, M., “On the stability of thermally stratified plane Poiseuille flow”, ZAMM 54 (1974) 533–540.CrossRefGoogle Scholar
[20]Vasilyev, O. V. and Paolucci, S., “Stability of unstably stratified shear flow in a channel under non-Boussinesq conditions”, Acta Mech. 112 (1995) 37–58.CrossRefGoogle Scholar