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An explanation for the multivalued heat transport found experimentally for convection in a porous medium

Published online by Cambridge University Press:  26 April 2006

C. R. B. Lister
C. R. B. Lister
Affiliation:
School of Oceanography, WB-10, University of Washington, Seattle, WA 98195, USA
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Abstract

Convection experiments were conducted in a porous slab 3 m in diameter and 30 cm thick, using two quite different media: a filling of rubberized curled coconut fibre and of clear polymethylmethacrylate beads. The second experiment involved the successful use of a visualization scheme for the flows at the upper boundary. Convection began in a hexagonal pattern with a slight tendency to form into rolls, but became very complex, irregular and three-dimensional at higher Rayleigh numbers, without developing any obvious temporal instabilities. Above a Rayleigh number of 1000 a significant number of dendritic downwellings appeared, where smaller downwellings seemed to feed into larger areas such that the whole complex may have converged into a single downwelling plume. The visualization provides direct confirmation that the lateral scale of the convection decreases with increasing Rayleigh number, approximately as ([Ascr ] + C)−0.5.

Nusselt number versus Rayleigh number curves were obtained for both experiments. The only feature they have in common is a central section where the slope on a log/log graph is slightly over 0.5. On the graph from the first experiment, this section is preceded by a slope close to 1 and followed by a slope close to 0.33. The temperature measured at a point in the fill 25 mm below the top boundary was unsteady at conditions representative of the upper two segments of the graph; sensitivity was insufficient to measure fluctuations at lower temperature differences. The Nusselt number for the bead fill jumps upward just above onset (where [Ascr ] = 4π2), rapidly settles to a slope of 0.52, and then gradually breaks upward again to a slope of greater than 1. Increases in conductivity and permeability close to the boundary are not a large enough fraction of the boundary-layer thickness to cause this. A new phenomenon, lateral thermal dispersion, appears to be responsible. It occurs because there is no constant separation distance between adjacent channels in a bead fill. Thermal exchange in the junction pores exceeds the average if the flow is fast enough, especially when the fluid is more conductive than the beads.

A simple boundary-conduction theory can be matched to the uncontaminated results. It is based on relative scaling of the residence time of fluid in the boundary layer, and predicts Nusselt number growth as the 0.55 power of Rayleigh number toward the high values typical of major geothermal areas.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 214 , May 1990 , pp. 287 - 320
Copyright
© 1990 Cambridge University Press

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References

Baker, D. J. 1966 A technique for the precise measurement of small fluid velocities. J. Fluid Mech. 26, 573–575.Google Scholar
Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76, 741–754.Google Scholar
Bories, S. 1970 Sur les mécanismes fondamenteaux de la convection naturelle en milieu poreux. Rev. Gen. Therm. 108, 1377–1401.Google Scholar
Bories, S. A. & Thirriot, C. 1969 Echanges thermiques et tourbillons dans une couche poreuse horizontale. Houille Blanche 3, 237–245.Google Scholar
Buretta, R. J. & Berman, A. S. 1976 Convective heat transfer in a liquid-saturated porous layer. Trans. ASME E: J. Appl. Mech. 43, 249–253.Google Scholar
Busse, F. & Joseph, D. D. 1972 Bounds for heat transport in a porous layer. J. Fluid Mech. 54, 521–543.Google Scholar
Caltagirone, J. P., Cloupeau, M. & Combarnous, M. A. 1971 Convection naturelle fluctuante dans une couche poreuse horizontale. C. R. Hebd. Séanc, Acad. Sci. Paris B, 833–836.Google Scholar
Combarnous, M. 1970 Convection naturelle et convection mixte dans une couche poreuse horizontale. Rev. Gen. Therm. 108, 1355–1375.Google Scholar
Combarnous, M. A. & Bia, P. 1971 Combined free and forced convection in porous media. Soc. Petrol. Engng J. 4, 399–405.Google Scholar
Combarnous, M. & Bories, S. 1974 Modélisation de la convection naturelle au sein d'une couche poreuse horizontale à l'aide d'un coefficient de transfert solide-fluide. Intl J. Heat Mass Transfer 17, 505–515.Google Scholar
Combarnous, M. A. & Bories, S. A. 1975 Hydrothermal convection in saturated porous media. In Advances in Hydroscience, vol. 10 (ed. V. T. Chow), pp. 231–307. Academic.
Corliss, J., Dymond, J., Gordon, L. I., Edmond, J. M., Herzen, R. P. von Ballard, R. D., Green, K., Williams, D., Bainbridge, A., Crane, K. & Van Andel, T. H. 1979 Submarine thermal springs on the Galapagos Rift. Science 203, 1073–1083.Google Scholar
Darcy, H. P. G. 1856 Les Fontaines Publiques de la Ville de Dijon. Paris: Victor Dalmont.
Davis, E. E. & Lister, C. R. B. 1974 Fundamentals of ridge-crest topography. Earth Planet. Sci. Lett. 21, 405–413.Google Scholar
Elder, J. W. 1965 Physical processes in geothermal areas. In Terrestrial Heat Flow (ed. W. H. K. Lee) Am. Geophys. U. Monog. no. 8, pp. 211–239.
Elder, J. W. 1967 Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 29–48.Google Scholar
Fehn, U. & Cathles, L. M. 1979 Hydrothermal convection at slow-spreading mid-ocean ridges. Tectonophys. 55, 239–260.Google Scholar
Fisher, R. G. 1964 Geothermal heat flow at Wairakei during 1958. N. Z. J. Geol. Geophys. 7, 172–184.Google Scholar
Gibson, K. M. 1980 The design and construction of an experiment to observe porous convection at high Rayleigh numbers. M.Sc. thesis, University of Washington, Seattle, WA.
Grindley, G. W. & Browne, P. R. L. 1975 Structural and hydrological factors controlling the permeabilities of some hot-water geothermal fields. Proc. 2nd UN Symp. on Development & Use of Geothermal Resources, vol. 1, pp. 377–386. Washington, DC: US Govt. Print. Off.
Gupta, V. P. & Joseph, D. D. 1973 Bounds for heat transport in a porous layer. J. Fluid Mech. 57, 491–514.Google Scholar
Hartline, B. K. 1978 Topographic forcing of thermal convection in a Hele-Shaw cell model of a porous medium. Ph.D. thesis, University of Washington, Seattle, WA.
Hartline, B. K. & Lister, C. R. B. 1977 Thermal convection in a Hele-Shaw cell. J. Fluid Mech. 79, 379–389.Google Scholar
Hartline, B. K. & Lister, C. R. B. 1981 Topographic forcing of supercritical convection in a porous medium such as the oceanic crust. Earth Planet. Sci. Lett. 55, 75–86.Google Scholar
Hele-Shaw, H. S. J. 1898 Trans. Inst. Naval Archit. 40, 21.
Howard, L. N. 1966 Convection at high Rayleigh number. Proc. 11th Intl Cong. Appl. Mech., Munich 1964 (ed. H. Gortier), pp. 1109–1115.
Katsaros, K. B., Liu, W. T., Businger, J. A. & Tillman, J. E. 1977 Heat transport and thermal structure in the interfacial boundary layer measured in an open tank of water in turbulent free convection. J. Fluid Mech. 83, 311–335.Google Scholar
Khurana, A. 1988 Rayleigh-Bénard experiment probes transition from chaos to turbulence. Phys. Today 41 (6), 17–21.Google Scholar
Krishnamurti, R. 1970a On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow. J. Fluid Mech. 42, 295–308.Google Scholar
Krishnamurti, R. 1970b On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309–321.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521.Google Scholar
Lister, C. R. B. 1974 On the penetration of water into hot rock. Geophys. J. R. Astron. Soc. 39, 465–509.Google Scholar
Lister, C. R. B. 1977 Estimators for heat flow and deep rock properties based on boundary-layer theory. Tectonophys. 37, 157–171.Google Scholar
Lister, C. R. B. 1984 The basic physics of water penetration into hot rock. In Hydrothermal Processes at Seafloor Spreading Centers, NATO Adv. Res. Inst. (ed. P. A. Rona. K. Bostrom, K. Smith & L. Laubier). Plenum, New York.
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153–161.Google Scholar
Ping Cheng, 1978 Heat transfer in geothermal systems. Adv. Heat Transfer 14, 1–105.Google Scholar
Ribando, R. J., Torrance, K. E. & Turcotte, D. L. 1976 Numerical models for hydrothermal circulation in the oceanic crust. J. Geophys. Res. 81, 3007–3012.Google Scholar
Richter, F. M. 1973 Convection and the large-scale circulation of the mantle. J. Geophys. Res. 78, 8735–8745.Google Scholar
Roberts, G. O. 1977 Fast viscous convection. Geophys. Astrophys. Fluid Dyn. 8, 197–233.Google Scholar
Roberts, G. O. 1979 Fast viscous Bénard convection. Geophys. Astrophys. Fluid Dyn. 12, 235–272.Google Scholar
Robinson, J. L. & O'Sullivan, M. J. 1976 A boundary-layer model of flow in a porous medium at high Rayleigh number. J. Fluid Mech. 75, 459–467.Google Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Scheidegger, A. E. 1974 The Physics of Flow through Porous Media, 3rd edn. University of Toronto Press, 353 pp.
Schneider, K. J. 1963 Investigation of the influence of free thermal convection on heat transfer through granular material. In. Porc. 11th Intl Cong. Refrig., Munich, Paper 11–4, pp. 247–254.Google Scholar
Sondergeld, C. H. & Turcotte, D. L. 1977 An experimental study of two-phase convection in a porous medium with applications to geological problems. J. Geophys. Res. 82, 2045–2053.Google Scholar
Straus, J. M. 1974 Large-amplitude convection in porous media. J. Fluid Mech. 64, 57–63.Google Scholar
Straus, J. M. & Schubert, G. 1977 Thermal convection of water in a porous medium: effects of temperature- and pressure-dependent thermodynamic and transport properties. J. Geophys. Res. 82, 325–333.Google Scholar
Ward, P. L. 1972 Microearthquakes: prospecting tool and possible hazard in the development of geothermal resources. Geothermics 1, 3–12.Google Scholar
Wooding, R. A. 1957 Steady state free convection of a liquid in a saturated permeable medium. J. Fluid Mech. 2, 273–285.Google Scholar