Skip to main content
SpringerLink
Log in

On the stability of the natural convection flow in a square cavity heated from the side

  • Published:
Applied Scientific Research Aims and scope Submit manuscript

Abstract

The stability of the steady laminar natural-convection flow of air (Prandtl number 0.71) and water (Prandtl number 7.0) in a square cavity is calculated by numerically solving the unsteady, two-dimensional Navier-Stokes equations. The cavity has a hot and cold vertical wall and either conducting or adiabatic horizontal walls. The flow looses its stability at a lower Rayleigh number in the case of conducting horizontal walls than in the case of adiabatic horizontal walls. The flow of water is more stable than the flow of air. Directly above the critical Rayleigh number the unsteady flow shows a single-frequency oscillation. Air in the case of adiabatic horizontal walls is an exception and shows two frequencies. The instabilities in the cavity seem to be related to well-known elementary instability mechanisms. In the case of conducting and adiabatic horizontal walls the instability seems to be related to a Rayleigh/Bénard and a Tollmien-Schlichting instability respectively. The second instability for air in the case of adiabatic horizontal walls seems to be related to an instability after a hydraulic jump.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K.H. Winters, Hopf bifurcation in the double-glazing problem with conducting boundaries, J. Heat Transfer 109 (1987) 894–898.

    Google Scholar 

  2. P. Le Quéré and T. Alziary de Roquefort, Transition to unsteady natural convection of air in vertical differentially heated cavities: influence of thermal boundary conditions on the horizontal walls. Proc. 8th Int. Heat Transfer Conf. San Francisco (1986) 1533–1538.

  3. P. Le Quéré and T. Alziary de Roquefort, On the existence of multiple periodic solutions of the Boussinesq equations. Mechanics of Fluids, C.R. Acad. Sci. Paris, 306, II (1988) 681–687.

    Google Scholar 

  4. P. Le Quéré and T. Alziary de Roquefort, Transition to unsteady natural convection of air in differentially heated vertical cavities. 4th Int. Conf. on Numerical Methods in Laminar and Turbulent Flow, Pineridge Press (1985) 841–852.

  5. S.V. Patankar and D.B. Spalding, A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows. Int. J. Heat Mass Transfer 15 (1972) 1787–1806.

    Google Scholar 

  6. D.D. Joseph and D.H. Sattinger, Bifurcating time periodic solutions and their stability. Arch. Rational Mech. Anal. 45 (1972) 79–109.

    Google Scholar 

  7. J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences 42, Springer (1983).

  8. B. Roux (ed), Proc. of the Gamm-workshop “Numerical simulation of oscillatory convection in low Pr fluids”. To appear in the series “Notes on Numerical Fluid Mechanics”, Vieweg (1989).

  9. R.A.W.M. Henkes and C.J. Hoogendoorn, Towards a benchmark solution for oscillating convection: Contribution of the heat-transfer group at Delft University. To appear in [8] (1989).

  10. A.E. Gill, The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26 (1966) 515–536.

    Google Scholar 

  11. J. Patterson and J. Imberger, Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100 (1980) 65–86.

    Google Scholar 

  12. R. Yewell, D. Poulikakos and A. Bejan, Transient natural convection experiments in shallow enclosures. J. Heat Transfer 104 (1982) 533–538.

    Google Scholar 

  13. J.M. Hyun, Transient buoyant convection of a contained fluid driven by the changes in the boundary temperatures. J. Appl. Mech. 52 (1985) 193–198.

    Google Scholar 

  14. P.G. Drazin and W.H. Reid, Hydrodynamic stability. Cambridge University Press (1981).

  15. G.N. Ivey, Experiments on transient natural convection in a cavity. J. Fluid Mech. 144 (1984) 389–401.

    Google Scholar 

  16. P.R. Nachtsheim, Stability of free-convection boundary-layer flows. NASA TN D-2089 (1963).

  17. S.E. Haaland and E.M. Sparrow, Wave instability of natural convection on inclined surfaces accounting for nonparallelism of the basic flow. J. Heat Transfer 96 (1973) 405–407.

    Google Scholar 

  18. K.L. Tzuoo, T.S. Chen and B.F. Armaly, Wave instability of natural convection flow on inclined surfaces. J. Heat Transfer 107 (1985) 107–111.

    Google Scholar 

  19. S.L. Lee, T.S. Chen and B.F. Armaly, Wave instability characteristics for the entire regime of mixed convection flow along vertical flat plates. Int. J. Heat Mass Transfer 30 (1987) 1743–1751.

    Google Scholar 

  20. R.A.W.M. Henkes, A.M. Lankhorst and C.J. Hoogendoorn, Structure of the laminar natural convection flow in a square cavity heated from the side for infinitely large Rayleigh number. Proc. ASME Winter Annual Meeting, Natural Convection in Enclosures, HTD-99 (1988) 9–16.

  21. A.E. Gill and A. Davey, Instabilities of a buoyancy-driven system. J. Fluid Mech. 35 (1969) 775–798.

    Google Scholar 

  22. Y. Jaluria and B. Gebhart, On transition mechanisms in vertical natural convection flow. J. Fluid Mech. 66 (1974) 309–337.

    Google Scholar 

  23. B. Gebhart and R. Mahajan, Characteristic disturbance frequency in vertical natural convection flow. Int. J. Heat Mass Transfer 18 (1975) 1143–1148.

    Google Scholar 

  24. B. Gebhart, Transient response and disturbance growth in vertical buoyancy-driven flows. J. Heat Transfer 110 (1988) 1166–1174.

    Google Scholar 

  25. E.R.G. Eckert and E. Soehnghen, Interferometric studies on the stability and transition to turbulence of a free-convection boundary layer. Proc. Gen. Disc. Heat Transfer, London, (1951) 321–323.

  26. K. Kitamura, M. Koike, I. Fukuoka and T. Saito, Large eddy structure and heat transfer of turbulent natural convection along a vertical flat plate. Int. J. Heat Mass Transfer 28 (1985) 837–850.

    Google Scholar 

  27. D.G. Briggs and D.N. Jones, Two-dimensional periodic natural convection in a rectangular enclosure of aspect ratio one. J. Heat Transfer 107 (1985) 850–854.

    Google Scholar 

  28. P. Le Quéré and F. Penot, Numerical and experimental investigation of the transition to unsteady natural convection of air in a vertical differentially heated cavity. Proc. ASME Winter Annual Meeting, Bifurcation Phenomena in Thermal Processes and Convection, HTD-94 (1987) 75–82.

Download references

Author information

Authors and Affiliations

  1. Department of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600 GA, Delft, The Netherlands

    R. A. W. M. Henkes & C. J. Hoogendoorn

Authors
  1. R. A. W. M. Henkes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. J. Hoogendoorn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henkes, R.A.W.M., Hoogendoorn, C.J. On the stability of the natural convection flow in a square cavity heated from the side. Appl. sci. Res. 47, 195–220 (1990). https://doi.org/10.1007/BF00418051

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00418051

Keywords

Search

Navigation