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Influence of Prandtl number on instability mechanisms and transition in a differentially heated square cavity

Published online by Cambridge University Press:  26 April 2006

R. J. A. Janssen  and
R. A. W. M. Henkes
R. J. A. Janssen
Affiliation:
Faculty of Applied Physics, J.M. Burgers Centre for Fluid Mechanics, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands
R. A. W. M. Henkes
Affiliation:
Faculty of Aerospace Engineering, J.M. Burgers Centre for Fluid Mechanics, Delft University of Technology, PO Box 5058, 2600 GB Delft, The Netherlands
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Abstract

The transition from laminar to turbulent of the natural-convection flow inside a square, differentially heated cavity with adiabatic horizontal walls is calculated, using the finite-volume method. The purpose of this study is firstly to determine the dependence of the laminar-turbulent transition on the Prandtl number and secondly to investigate the physical mechanisms responsible for the bifurcations observed. It is found that in the square cavity, for Prandtl numbers between 0.25 and 2.0, the transition occurs through periodic and quasi-periodic flow regimes. One of the bifurcations is related to an instability occurring in a jet-like fluid layer exiting from those corners of the cavity where the vertical boundary layers are turned horizontal. This instability is mainly shear-driven and the visualization of the perturbations shows the occurrence of vorticity concentrations which are very similar to Kelvin–Helmholtz vortices in a plane jet, suggesting that the instability is a Kelvin–Helmholtz-type instability. The other bifurcation for Prandtl numbers between 0.25 and 2.0 occurs in the boundary layers along the vertical walls. It differs however from the related instability in the natural-convection boundary layer along an isolated vertical plate: the instability in the cavity is shear-driven whereas the instability along the vertical plate is mainly buoyancy-driven. For Prandtl numbers between 2.5 and 7.0, it is found that there occurs an immediate transition from the steady to the chaotic flow regime without intermediate regimes. This transition is also caused by instabilities originating and concentrated in the vertical boundary layers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 290 , 10 May 1995 , pp. 319 - 344
Copyright
© 1995 Cambridge University Press

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References

Blythe, P. A., Daniels, P. G. & Simpkins, P. G. 1983 Thermal convection in a cavity: the core structure near the horizontal boundaries. Proc. R. Soc. Lond. A 387, 367–388.Google Scholar
Comte, P., Lesieur, M. & Chollet, J. P. 1987 Simulation numérique d'un jet plan turbulent. C.R. Acad. Sci. Paris 305 Série II, 1037–1044.
De Vahl Davis, G. & Jones, I. P. 1983 Natural-convection of air in a square cavity. Intl J. Num. Meth. Fluids 3, 229–248.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Elder, J. W. 1965 Laminar free convection in a vertical slot. J. Fluid Mech. 23, 77–97.Google Scholar
Garg, V. K. 1981 Spatial stability of the non-parallel Bickley jet. J. Fluid Mech. 102, 127–140.Google Scholar
Gebhart, B., Jaluria, Y., Mahajan, R. L. and Sammakia, B. 1988 Buoyancy-Induced Flows and Transport. Hemisphere Publishing Corporation.
Gebhart, B. & Mahajan, R. 1975 Characteristic disturbance frequency in vertical natural convection flow. Intl J. Heat Mass Transfer 18, 1143–1148.Google Scholar
Gill, A. E. 1966 The boundary-layer regime for convection in a rectangular. J. Fluid Mech. 26, 515–536.Google Scholar
Gill, A. E. & Davey, A. 1969 Instabilities of a buoyancy-driven system. J. Fluid Mech. 35, 775–798.Google Scholar
Graebel, W. P. 1981 The influence of Prandtl number on free convection in a rectangular cavity. Intl J. Heat Mass Transfer 24, 125–131.Google Scholar
Henkes, R. A. W. M. 1990 Natural-convection boundary layers. PhD thesis, Delft University of Technology.
Henkes, R. A. W. M. & Hoogendoorn, C. J. 1993 Scaling of the laminar natural-convection flow in a heated square cavity. Intl J. Heat Mass Transfer 36, 2913–2925.Google Scholar
Ivey, G. N. 1984 Experiments on transient natural convection in a cavity. J. Fluid Mech. 144, 389–401.Google Scholar
Janssen, R. J. A. & Henkes, R. A. W. M. 1993 Accuracy of the finite-volume discretization for the bifurcating natural-convection flow in a heated square cavity. Numer. Heat Transfer B 24, 191–207.Google Scholar
Le Quéré, P. & Alziary De Roquefort, T. 1985 Transition to unsteady natural convection of air in differentially heated vertical cavities. 4th Intl Conf. Num. Meth. in Laminar and Turbulent Flow, pp. 841–852. Pineridge Press.
Le Quéré, P. & Alziary De Roquefort, T. 1986 Transition to unsteady natural convection of air in differentially heated vertical cavities: influence of thermal boundary condition on the horizontal walls. Proc. 8th Intl Heat Transfer Conf., San Francisco, pp. 1533–1538.
Meijerink, J. A. & Van Der Vorst, H. A. 1977 An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31, 148–162.Google Scholar
Nachtsheim, P. R. 1963 Stability of free-convection boundary-layer flows. NASA TN D-2089.
Paolucci, S. & Chenoweth, D. R. 1989 Transition to chaos in a differentially heated vertical cavity. J. Fluid Mech. 201, 379–410.Google Scholar
Patterson, J. C. & Armfield, S. W. 1992 Transient features of natural convection in a cavity. J. Fluid Mech. 219, 469–498.Google Scholar
Patterson, J. C. & Imberger, J. 1980 Unsteady natural convection in a square cavity. J. Fluid Mech. 100, 65–86.Google Scholar
Ravi, M. R., Henkes, R. A. W. M. & Hoogendoorn, C. J. 1994 On the high Rayleigh number structure of steady laminar natural-convection flow in a square enclosure. J. Fluid Mech. 262, 325–351.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192.Google Scholar
Schladow, S. G. 1990 Oscillatory motion in a side-heated cavity. J. Fluid Mech. 213, 589–610.Google Scholar
Schladow, S. G., Patterson, J. C. & Street, R. L. 1989 Transient flow in a side-heated cavity at high Rayleigh number: a numerical study. J. Fluid Mech. 200, 121–148.Google Scholar
Schlichting, H. 1968 Boundary-Layer Theory. McGraw-Hill.
Thorpe, S. A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32, 489–528.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Van Der Vorst, H. A. 1989 High performance preconditioning. SIAM J. Sci. Stat. Comput. 10, 1174–1185.Google Scholar
Winters, K. H. 1987 Hopf bifurcation in the double-glazing problem with conducting boundaries. J. Heat Transfer 109, 894–898.Google Scholar
Winters, K. H. 1988 Oscillatory convection in liquid metals in a horizontal temperature gradient. Intl J. Numer. Methods Engng 25, 401–414.Google Scholar
Winters, K. H. 1989 A bifurcation analysis of oscillatory convection in liquid metals. Proc. Gamm-Workshop on Numerical Simulation of Oscillatory Convection in Low Pr Fluids, pp. 319–326. Vieweg.