Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-29T05:08:47.626Z Has data issue: false hasContentIssue false

Vortex–wave interactions: long-wavelength streaks and spatial localization in natural convection

Published online by Cambridge University Press:  12 June 2012

Philip Hall*
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, UK
*
Email address for correspondence: philhall@ic.ac.uk

Abstract

The unidirectional shear flow driven by buoyancy effects in a vertical channel when a temperature difference is maintained between the walls of the channel is considered. The flow is unstable to waves which interact to reinforce the original flow and make it ‘streaky’. Such ‘vortex–wave’ interactions have been the subject of much recent research but little is yet known about what happens when the wavelength of the roll/streak flow becomes large. An asymptotic structure for long-wavelength interactions is derived and the tendency of the fluid to resist this state and the flow to become localized is revealed. Here the high-Grashof-number limit is considered and it is shown how a self-sustained process can occur with vortices interacting with a wave system in a manner similar to that discussed by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The work is closely related to numerical simulations of self-sustained processes in for example Couette flow but the fact that the basic flow here is generated by buoyancy effects enables us to make analytical progress. It is shown that the wave part of the interaction process has a flat critical layer and its wavelength is twice that of the streaky flow which supports it. Such subharmonic vortex–wave/self-sustained process interactions have not been previously identified.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

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

1. Benney, D. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Maths 70, 119.CrossRefGoogle Scholar
2. Bergholz, R. F. 1978 Instability of natural convection in a vertical fluid layer. J. Fluid Mech. 84, 743768.CrossRefGoogle Scholar
3. Bratsun, D., Zyuzgin, A. V. & Putin, G. F. 2003 Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side. Intl J. Heat Transfer Fluid Flow 24, 835852.Google Scholar
4. Fedorovich, E. & Shapiro, A. 2009 Turbulent natural convection along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech. 636, 4157.CrossRefGoogle Scholar
5. Gibson, J., Halcrow, J. & Cvitanovic, P. F. 2008 Visualising the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.Google Scholar
6. Gotoh, K & Ikeda, N. 1972 Secondary convection in a fluid between parallel vertical plates of different temperatures. J. Phys. Soc. Japan 661, 178205.Google Scholar
7. Hall, P. 2012 Vortex–wave interactions in natural convection in the high Prandtl number limit. Stud. Appl. Maths (in press).Google Scholar
8. Hall, P. & Horseman, N. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357375.CrossRefGoogle Scholar
9. Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.CrossRefGoogle Scholar
10. Hall, P. & Smith, F. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641666.CrossRefGoogle Scholar
11. Ruth, D. W. 1979 On the transition to transverse rolls in an infinite vertical fluid layer-a power series solution. Intl J. Heat Mass Transfer 22, 11991208.CrossRefGoogle Scholar
12. Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 15171534.CrossRefGoogle Scholar
13. Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.CrossRefGoogle ScholarPubMed
14. Willis, A. P. & Kerswell, R. R. 2009 Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localized ‘edge’ states. J. Fluid Mech. 619, 213233.CrossRefGoogle Scholar