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Modal exchange mechanisms in Lapwood convection

Published online by Cambridge University Press:  26 April 2006

D. S. Riley  and
K. H. Winters
D. S. Riley
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
K. H. Winters
Affiliation:
Theoretical Physics Division, Harwell Laboratory, Didcot, Oxon OX11 0RA, UK
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Abstract

Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Bénard problem: Lapwood convection in a two-dimensional saturated porous cavity heated from below. Two particular aspects of the problem are focused upon: (i) the existence of multiple steady solutions and (ii) the influence of aspect ratio.

Convection begins only when the applied temperature difference (say) exceeds a critical value defined by linear stability theory. The resulting convective flow pattern depends both on the magnitude of the temperature difference and on the aspect ratio of the cavity. A weakly nonlinear analysis reveals the roles played by so-called secondary bifurcations in determining the formation of further, anomalous patterns at fixed aspect ratio. In addition to giving rise to alternative stable flows for identical operating conditions, the secondary bifurcations are required for the modal exchanges which take place as the aspect ratio varies, a process which causes an abrupt change in preferred flow pattern at certain critical values of the aspect ratio.

As a complement to and an extension of the weakly nonlinear analysis, numerical methods are used to determine the bifurcation processes and to elucidate the modal exchange mechanisms in both weakly and strongly convective flows. The effect of container size is studied by continuation methods to predict the variation of the critical Rayleigh number of the bifurcation points for aspect ratios in the range 0.5 to 2.0. In this way a stability map is obtained which shows the alternative patterns expected for particular operating conditions. The Nusselt number is computed and it is found that the alternative stable modes transfer significantly different amounts of heat through the medium.

The study has provided new information on the existence and characteristics of, and interactions between, alternative steady modes of two-dimensional Lapwood convection. The results have important ramifications for the modelling and design of physical systems in which convective flow in a saturated porous medium is stimulated by an imposed unstable temperature gradient.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 325 - 358
Copyright
© 1989 Cambridge University Press

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References

Aidun, C. K. & Steen, P. H., 1987 Transition to oscillatory convective heat transfer in fluidsaturated porous medium. J. Thermophys. 1, 268.Google Scholar
Bauer, L., Keller, H. B. & Reiss, E. L., 1975 Multiple eigenvalues lead to secondary bifurcation. SIAM Rev. 17, 101.Google Scholar
Beck, J. L.: 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377.Google Scholar
Bories, S. A.: 1985 Natural convection in porous media. Proc. NATO ASI on Fundamentals of Transport Phenomena in Porous Media.Google Scholar
Bories, S. A., Combarnous, M. A. & Jaffrennou, J. Y., 1972 Observation des différentes formes d'écoulements convectifs dans une couche poreuse inclinée. C. R. Acad. Sci. Paris A 275, 857.Google Scholar
Borkowska-Pawlak, B. & Kordylewski, W. 1982 Stability of two-dimensional natural convection in a porous layer. Q. J. Mech. Appl. Maths, 35, 279.Google Scholar
Borkowska-Pawlak, B. & Kordylewski, W. 1985 Cell-pattern sensitivity to box configuration in a saturated porous media. J. Fluid Mech. 150, 169.Google Scholar
Caltagirone, J. P.: 1975 Thermoconvective instabilities in a horizontal porous layer. J. Fluid Mech. 72, 269.Google Scholar
Caltagirone, J. P., Cloupeau, M. & Combarnous, M. A., 1971 Convection naturelle fluctuante dans une couche poreuse horizontale. C. R. Acad. Sci. Paris B 273, 833.Google Scholar
Caltagirone, J. P., Meyer, G. & Mojtabi, A., 1981 Structurations thermoconvectives tridimensionelles dans une couche poreuse horizontale. J. Méc. 20, 219.Google Scholar
Cliffe, K. A.: 1983 Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219.Google Scholar
Cliffe, K. A.: 1988 Numerical calculations of the primary flow exchange process in the Taylor Problem. J. Fluid Mech. 197, 57.Google Scholar
Cliffe, K. A. & Winters, K. H., 1984 A numerical study of the cusp catastrophe for Bénard convection in tilted cavities. J. Comp. Phys. 54, 531.Google Scholar
Cliffe, K. A. & Winters, K. H., 1986 The use of symmetry in bifurcation calculations and its application to the Bénard problem. J. Comp. Phys. 67, 310.Google Scholar
Combarnous, M. A. & LeFur, B., 1969 Transfert de chaleur par convection naturelle dans une couche poreuse horizontale. C. R. Acad. Sci. Paris B 269, 1009.Google Scholar
Golubitsky, M. & Schaeffer, D. G., 1985 Singularities and Groups in Bifurcation Theory, Part I. Springer.
Holst, P. H. & Aziz, K., 1972 Transient three-dimensional natural convection in confined porous media. Intl J. Heat Mass Transfer 15, 73.Google Scholar
Horne, R. N.: 1979 Three-dimensional natural convection in a confined porous medium heated from below. J. Fluid Mech. 92, 751.Google Scholar
Horne, R. N. & O'Sullivan, M. J. 1974 Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339.Google Scholar
Jackson, C. P.: 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23.Google Scholar
Jackson, C. P. & Winters, K. H., 1984 A finite-element study of the Bénard problem using parameter-stepping and bifurcation search. Intl J. Num. Meth. Fluids 4, 127.Google Scholar
Jepson, A. D. & Spence, A., 1984 Singular points and their computation. In Numerical Methods for Bifurcation Problems (ed. T. Küpper, H. D. Mittleman & H. Weber). International Series of Numerical Mathematics, Vol. 70. Birkäuser.
Jepson, A. D., Spence, A. & Cliffe, K. A., 1987 The numerical solution of nonlinear equations having several parameters. Part III: equations with Z2-symmetry. Harwell Rep. TP. 1210 and submitted to SIAM J. Num. Anal.Google Scholar
Keller, H. B.: 1977 Numerical solutions of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), pp. 359–384. Academic.
Kidachi, H.: 1982 Side wall effect on the pattern formation of the Rayleigh–Bénard convection. Prog. Theor. Phys. 68, 49.Google Scholar
Kimura, S., Schubert, G. & Straus, J. M., 1987 Instabilities of steady, periodic and quasiperiodic modes of convection in porous media. Trans. ASME C: J. Heat Transfer 109, 350.Google Scholar
Metzener, P.: 1986 The effect of rigid sidewalls on nonlinear two-dimensional Bénard convection. Phys. Fluids 29, 1373.Google Scholar
Moore, G. & Spence, A., 1980 The calculation of limit points of non-linear equations. SIAM J. Numer. Anal. 17, 567.Google Scholar
Rees, D. A. S. & Riley, D. S. 1986 Free convection in an undulating saturated porous layer: resonant wavelength excitation. J. Fluid Mech. 166, 503.Google Scholar
Rees, D. A. S. & Riley, D. S. 1989a The effects of boundary imperfections on convection in a saturated porous layer: near-resonant wavelength excitation. J. Fluid Mech. 199, 133.Google Scholar
Rees, D. A. S. and Riley, D. S.: 1989b The effects of boundary imperfections on convection in a saturated porous layer: non-resonant wavelength excitation. Phil. Trans. R. Soc. Lond. (Submitted.)Google Scholar
Riley, D. S. & Winters, K. H., 1986 The onset of convection in a porous medium: a preliminary study. Harwell Rep. R.12586.Google Scholar
Riley, D. S. & Winters, K. H., 1987 A bifurcation study of convection in a two-dimensional saturated porous cavity. Harwell Rep. TP.1246 and Bifurcation Phenomena in Thermal Processes and Convection, HTD, Vol. 94 and AMD, Vol. 89. ASME.
Schaeffer, D. G.: 1980 Qualitative analysis of a model for boundary effects in the Taylor problem. Proc. Camb. Phil. Soc. 87, 307.Google Scholar
Shearer, M.: 1980 Secondary bifurcation near a double eigenvalue SIAM J. Math. Anal. 11, 365.
Steen, P.: 1983 Pattern selection for finite-amplitude convection states in boxes of porous media. J. Fluid Mech. 136, 219.Google Scholar
Steen, P.: 1985 Container geometry and the transition to unsteady Bénard convection in porous media. Phys. Fluids 29, 925.Google Scholar
Schubert, G. & Straus, J. M., 1979 Three-dimensional and multi-cellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers. J. Fluid Mech. 94, 25.Google Scholar
Straus, J. M.: 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 51.Google Scholar
Straus, J. M. & Schubert, G., 1978 On the existence of three-dimensional convection in a rectangular box containing fluid-saturated porous material. J. Fluid Mech. 87, 385.Google Scholar
Straus, J. M. & Schubert, G., 1979 Three-dimensional convection in a cubic box of fluidsaturated porous material. J. Fluid Mech. 91, 155.Google Scholar
Straus, J. M. & Schubert, G., 1981 Modes of finite-amplitude three-dimensional convection in rectangular boxes of fluid-saturated porous material. J. Fluid Mech. 103, 23.Google Scholar
Sutton, F.: 1970 Onset of convection in a porous channel with net through flow. Phys. Fluids 13, 1931.Google Scholar
Tavener, S. & Cliffe, K. A., 1988 Primary flow exchange mechanisms in Taylor–Couette flow applying non-flux boundary conditions. Harwell Rep. TP.1316.Google Scholar
Werner, B.: 1984 Regular systems for bifurcation points with underlying symmetries. In Numerical Methods for Bifurcation Problems (ed. T. Küpper, H. D. Mittleman & H. Weber). International Series of Numerical Mathematics, Vol. 70. Birkhäuser.
Winters, K. H.: 1987a Hopf bifurcation in the double-glazing problem. Harwell Rep. TP.1187 and Trans. ASME C: J. Heat Transfer 109, 894–898.
Winters, K. H.: 1987b A bifurcation study of laminar flow in a curved tube of rectangular crosssection. J. Fluid Mech. 180, 343.Google Scholar
Winters, K. H.: 1987c Oscillatory convection in crystal melts: the horizontal Bridgman process. In Proc. 5th Intl Conf on Numerical Methods in Thermal Problems, Montreal, pp. 299–310. Swansea: Pineridge.
Winters, K. H. & Brindley, R. C. G. 1984 Multiple solutions for laminar flow in helically-coiled tubes. Harwell Rep. R.11373.Google Scholar
Winters, K. H. & Brown, T. W., 1985 Bénard convection in a three-dimensional cavity: the effects of aspect ratio and tilt. In Proc. 4th Intl Conf. on Numerical Methods in Thermal Problems, Swansea, pp. 376–385. Swansea: Pineridge.
Winters, K. H. & Cliffe, K. A., 1985 The prediction of critical points for thermal explosions in a finite volume. Combust. Flame 62, 13.Google Scholar
Winters, K. H., Cliffe, K. A. & Jackson, C. P., 1984 A review of extended systems for finding critical points in coupled problems. In Proc. Conf. on Numerical Methods for Transient and Coupled Problems (ed. R. W. Lewis, E. Hinton, P. Bettess & B. A. Schrefler), pp. 949–959. Swansea: Pineridge.
Winters, K. H., Cliffe, K. A. & Jackson, C. P., 1987 The prediction of instabilities using bifurcation theory. Numerical Methods for Transient and Coupled Problems (ed. R. W. Lewis, E. Hinton, P. Bettess & B. A. Schrefler), pp. 179–198. Wiley.
Winers, K. H., Plesser, Th. & Cliffe, K. A., 1988 The onset of convection in a finite container due to surface tension and buoyancy. Physica 29D, 387–401.Google Scholar