Skip to main content Accessibility help
×
Home

Neutrally stable wave motions in thermally stratified Poiseuille-Couette flow

  • James P. Denier (a1) and Andrew P. Bassom (a2)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Neutrally stable wave motions in thermally stratified Poiseuille-Couette flow
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Neutrally stable wave motions in thermally stratified Poiseuille-Couette flow
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Neutrally stable wave motions in thermally stratified Poiseuille-Couette flow
      Available formats
      ×

Copyright

References

Hide All
[1]Abramowitz, M. and Stegun, I. A. (eds.). Handbook of Mathematical Functions (Dover, New York, 1965).
[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) 701710.
[3]Cowley, S. J. and Smith, F. T., “On the stability of Poiseuille–Couette flow: a bifurcation from infinity”, J. Fluid Mech. 156 (1985) 83100.
[4]Deardorff, J. W., “Gravitational instability between horizontal plates with shear”, Phys. Fluids 8 (1965) 10271030.
[5]Denier, J. P., “Nonlinear wave interactions in stratified Poiseuille–Couette flow” (1998), in preparation.
[6]Denier, J. P. and Mureithi, E. W., “Weakly nonlinear wave motions in a thermally stratified boundary layer”, J. Fluid Mech. 315 (1996) 293316.
[7]Drazin, P. G. and Reid, W. H., Hydrodynamic stability (C.U.P., 1979).
[8]Fujimura, K. and Kelly, R. E., “Stability of unstably stratified shear flow between parallel plates”, Fluid Dyn. Res. 2 (1988) 281292.
[9]Gage, K. S., “The effect of stable thermal stratification on the stability of viscous parallel flows”, J. Fluid Mech. 47 (1974) 120.
[10]Gage, K. S. and Reid, W. H., “The stability of thermally stratified plane Poiseuille flow”, J. Fluid Mech. 33 (1968) 2132.
[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) 117128.
[12]Hall, P., “Taylor-Gortler vortices in fully developed or boundary layer flows: linear theory”, J. Fluid Mech. 124 (1982) 475494.
[13]Hughes, T. H. and Reid, W. H., “The stability of spiral flow between rotating cylinders”, Phil. Trans. R. Soc. Lond. 263 (1968) 5791.
[14]Koppel, D., “On the stability of flow of a thermally stratified fluid under the action of gravity”, J. Math. Phys. 5 (1964) 963982.
[15]Miles, J. W., “On the stability of heterogeneous shear flows”, J. Fluid Mech. 10 (1961) 496508.
[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) 1950.
[17]Otto, S. R. and Bassom, A. P., “Weakly nonlinear stability of viscous vortices in three-dimensional boundary layers”, J. Fluid Mech. 249 (1993) 597618.
[18]Schäfer, P. and Herwig, H., “Stability of plane Poiseuille flow with temperature dependent viscosity”, Int. J. Heat Mass Transfer 36 (1993) 24412448.
[19]Tveitereid, M., “On the stability of thermally stratified plane Poiseuille flow”, ZAMM 54 (1974) 533540.
[20]Vasilyev, O. V. and Paolucci, S., “Stability of unstably stratified shear flow in a channel under non-Boussinesq conditions”, Acta Mech. 112 (1995) 3758.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO

Neutrally stable wave motions in thermally stratified Poiseuille-Couette flow

  • James P. Denier (a1) and Andrew P. Bassom (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.