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Convection and flow in porous media. Part 1. Visualization by magnetic resonance imaging

  • M. D. Shattuck (a1), R. P. Behringer (a1), G. A. Johnson (a2) and J. G. Georgiadis (a3)
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We describe an experimental study of porous media convection (PMC) from onset to 8Rac. The goal of this work is to provide non-invasive imaging and high-precision heat transport measurements to test theories of convection in PMC. We obtain velocity information and visualize the convection patterns using magnetic resonance imaging (MRI). We study both ordered and disordered packings of mono-disperse spheres of diameter d = 3.204 ± 0.029 mm, in circular, rectangular, and hexagonal planforms. In general, the structure of the medium plays a role which is not predicted by theories which assume a homogeneous system. Disordered media are prepared by pouring mono-disperse spheres into the container. Large ordered regions of close packing for the spheres, with grain boundaries and isolated defects, characterize these media. The defects and grain boundaries play an important role in pattern formation in disordered media. Any deviation from close packing produces a region of larger porosity, hence locally larger permeability. The result is spatial variations in the Rayleigh number, Ra. We define the critical Ra, Rac , as the Rayleigh number at the onset of convection in the ordered regions. We find that stable localized convective regions exist around grain boundaries and defects at Ra < Rac. These remain as pinning sites for the convection patterns in the ordered regions as Ra increases above Rac up to 5Rac , the highest Ra studied in the disordered media. In ordered media, spheres are packed such that the only deviations from close packing occur within a thin (<d) region near the vertical walls. Stable localized convection begins at 0.5Rac in the wall regions but appears to play only a weak role in the pattern formation of the interior regions (bulk), since different stable patterns are observed in the bulk at the same Ra after each cycling of Ra below Rac , even for similar patterns of small rolls in the wall regions. The experiments provide a test of the following predictions for PMC: (i) that straight parallel rolls should be linearly stable for Rac < Ra < 5Rac; (ii) that at onset, the rolls should have a dimensionless wavevector qc = π; (iii) that at the upper end of this range rolls should lose stability to cross-rolls; (iv) that the initial slope of the Nusselt curve should be 2; (v) that there should be a rapid decay of vertical vorticity - hence no complex flows, such as those which occur for Rayleigh- Benard convection (RBC) within the nominal regime of stable parallel rolls. These predictions are in partial agreement with our findings for the bulk convection in the ordered media. We observe roll-like structures which relax rapidly to stable patterns between Rac and 5Rac. However we find a wavenumber which is 0.7π compared to π derived from linear stability theory. We find an asymmetry between the size of the upfiowing regions and downfiowing regions as Ra grows above Rac. The ratio of the volume of the upfiowing to the volume of the downfiowing regions decreases as Ra increases and leads to a novel time-dependent state, which does not consist of cross-rolls. This time-dependent state begins at 6Rac and is observed up to 8Rac , the largest Ra which we studied. It seems likely that the occurrence of this state is linked to departures from the Boussinesq approximation at higher Ra. We also find that the slope of the Nusselt curve is 0.7, which does not agree with the predicted value of 2.

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Ashcroft, N. W. & Mermin, N. D. 1976 Solid State Physics. Fort Worth: Saunders.
Bear, J. 1988 Dynamics of Fluids in Porous Media. Dover.
Beckermann, C., Viskanta, R. & Ramadhyani, S. 1988 Natural convection in vertical enclosuers containing simultaneously fluid and porous layers. J. Fluid Mech. 186, 257284.
Behringer, R. P. 1985 Rayleigh-Benard convection and turbulence in liquid helium. Rev. Mod. Phys. 57, 657687.
Bloch, F. 1946 Nuclear induction. Phys. Rev. 70, 460474.
Bodenschatz, E., Bruyn, J. De, Ahlers, G. & Cannell, D. S. 1991 Transitions between patterns in thermal convection. Phys. Rev. Lett. 67, 30783081.
Bories, S. & Thirriot, C. 1969 E'changes thermiques et tourbillons dans une couche poreuse horizontale. La Houille Blanche 24, 237245.
Boussinesq, J. 1903 Theorie Analytique de la Chaleur, vol. 2. Gauthier-Villars.
Bruyn, J. De, Bodenschatz, E., Morris, S. W., Trainoff, S. P., Hu, Y., Cannell. D. S. & Ahlers, G. 1996 Apparatus for the study of Rayleigh-Benard convection in gas under pressure. Rev. Sci. Instrum. 67, 20432067.
Buretta, R. J. & Berman, A. S. 1976 Convective heat transfer in a liquid saturated porous layer. Trans. ASME J. Appl. Mech. 43, 249253.
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625649.
Busse, F. H. 1985 Transition to turbulence in Rayleigh-Benard convection. In Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn (ed. Swinney, H. L. & Gollub, J.), pp. 97137. Springer.
Busse, F. H. & Joseph, D. D. 1972 Bounds for heat transport in a porous layer. J. Fluid Mech. 54, 521543.
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305320.
Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy. Clarendon.
Close, D. J., Symons, J. G. & White, R. F. 1985 Convective heat tranfer in shallow gas-filled porous media. Ml J. Heat Mass Transfer 28, 23712378.
Combarnous, M. 1970 Convection naturelle et convection mixte dans une couche poreuse horizontale. Rev. Gen. Therm. 9, 13551375.
Combarnous, M. & BORIES, S. A. 1975 Hydrothermal convection in saturated porous media. Hydroscience 10, 231307.
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 8511112.
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Dalmont.
Dullien, F. A. L. 1979 Porous Media: Fluid Transport and Pore Structure. Academic.
Dumoulin, C. L. 1986 Magnetic resonance angiography. Radiology 161, 717720.
Elder, J. W. Steady free convection in a porous medium heated from below. J. Fluid Mech. 11, 2948.
Ene, H. I. & Polisevski, D. 1987 Thermal Flow in Porous Media. Reidel.
Ergun, S. 1952 Fluid flow through packed columns. Chem. Engng Prog. 48, 8994.
Georgiadis, J. G. 1991 Future research needs in convective heat and mass transport in porous media. In Convective Heat and Mass Transfer (ed. Kakac, S., Kilkis, B., Kulacki, F. A. & Arinc, F.). NATO ASI Series E, vol. 196, pp. 10731088. Kluwer.
Georgiadis, J. G., Behringer, R. P., Shattuck, M. D. & Johnson, G. A. 1991 Interstitial velocity and temperature fields in fully-saturated porous media. In Proc. 9th Symp. on Energy Sciences. US Department of Energy.
Givler, R. C. & Altobelli, S. A. 1994 A determination of the effective viscosity for the Brinkman- Forchheimer flow model. J. Fluid Mech. 258, 355370.
Greenside, H. S., Cross, M. C. & Coughran, W. M. 1988 Mean flows and the onset of chaos in large-cell convection. Phys. Rev. Lett. 60, 22692272.
Gupta, V. P. & Joseph, D. D. 1973 Bounds for heat transport in a porous layer. J. Fluid Mech. 57. 491514.
Horton, C. W. & Rodgers, F. T. 1945 Convection currents in a porous media. J. Appl. Phys. 16, 367370.
Howle, L. E., Behringer, R. P. & Georgiadis, J. G. 1993a Shadowgraphic visualization of natural convection in rectangular-grid porous layers. In Topics in Heat Transfer I (ed. Keyhani, M. et al). HTD, vol. 206, pp. 230232.
Howle, L. E., Behringer, R. P. & Georgiadis, J. G. 1993b Visualization of convective fluid flow in a porous medium. Nature 362, 230232.
Howle, L. E., Behringer, R. P. & Georgiadis, J. G. 1997 Convection in ordered and disordered porous media. Part 2. Visualization by shadowgraph. J. Fluid Mech. 332, 247262.
Joseph, D. D. 1976 Stability of Fluid Motions, II. Springer.
Kaneko, T., Mohtadi, M. F. & Aziz, K. 1974 An experimental study of natural convection in inclined porous media. Intl J. Heat Mass Transfer 17, 485–196.
Katto, Y. & Masuoka, T. 1967 Criterion for the onset of convective flow in a fluid in a porous medium. Intl J. Heat Mass Transfer 10, 297309.
Kutsovsky, Y. E., Scriven, E., Davis, H. T. & Hammer, B. E. 1996 NMR image of velocity profiles and velocity distributions in bead packs. Phys. Fluids 8, 863871.
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508521
Lebon, L., Oger, L., Leblond, J., Hulin, J. P., Martys, N. S. & Schwartz, L. M. 1996 Pulsed gradkiend NMR measurements and numerical simulation of flow velocity distribution in sphere packings. Phys. Fluids 8, 293301.
Lein, H. & Tankin, R. S. 1992 Natural convection in porous media - I. Nonfreezing. Intl J. Heat Mass Transfer 35, 175186.
Lister, C. R. B. 1990 An explanation for the multivalued heat transport found experimentally for convection in a porous medium. J. Fluid Mech. 214, 287320.
Martys, N. S., Torquato, T. & Benz, T. P. 1994 Universal scaling of fluid permeability for sphere packings. Phys. Rev. E 50, 403408.
Moran, P. R. 1982 A flow velocity zeugmatographic interlace for NMR imaging in humans. Magnetic Resonance Imaging 1, 197203.
Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. 1993 Spiral defect chaos in large aspect ratio Rayleigh-Benard convection. Phys. Rev. Lett. 71, 20262029.
Newell, A. C., Passot, T. & Lega, J. 1993 Order parameter equations for patterns. Ann. Rev. Fluid Mech. 25, 399453.
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279303.
Nield, D. A. & Bejan, A. 1992 Convection in Porous Media. Springer.
Oberbeck, A. Ueber die Warmeleitung der Flussigkeiten bei Beriicksichtigung der Stromungen infolge con Temperaturdifferenzen. Ann. Phys. Chem. 7, 271292.
O'Donnell, M. 1985 NMR blood flow imaging using multiecho, phase contrast sequences. Med. Phys. 12, 5964.
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153161.
Prasad, V., Kulacki, F. A. & Keyhani, M. 1985 Natural convection in porous media. J. Fluid Mech. 150, 89119.
Rasenat, S., Hartung, G., Winkler, B. L. & Rehberg, I. 1989 The shadowgraph method in convection experiments. Exps. Fluids 7, 412420.
Redpath, T. W., Norris, D. G., Jones, R. A. & Hutchison, J. M. S. 1984 A new method of NMR flow imaging. Phys. Med. Biol. 29, 891898.
Schneider, K. J. 1963 Investigation on the influence of free thermal convection on the heat transfer through granular material. Proc. 11th Intl Congr. of Refrigeration, pp. 247253. Pergamon.
Sigel, L. A. 1969 Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203224.
Shattuck, M. 1995 Flows in porous media: visualization by magnetic resonance imaging. Unpublished: Duke University.
Shattuck, M. D., Behringer, R. P., Johnson, G. A. & Georgiadis, J. G. 1994 Magnetic resonance imaging of convection in porous media. In Proc. 12th Symp. on Energy Sciences.
Shattuck, M. D., Behringer, R. P., Johnson, G. A. & Georgiadis, J. G. 1995 Onset and stability of convection in porous media: visualization by magnetic resonsnce imaging. Phys. Rev. Lett. 75, 19341937.
Siggia, E. D. & Zippelius, A. 1981 Pattern selection in Rayleigh-Benard convection near threshold. Phys. Rev. Lett. 47, 835838.
Singer, J. R. 1980 Blood flow measurements by NMR of the intact body. IEEE Trans. Nuclear Sci. 27, 12451249.
Slichter, C. P. 1978 Principles of Magnetic Resonance. Springer.
Straus, J. M. 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 5163.
Torrance, K. E., Schoenhals, R. J., Tien, C. L. & Viskanta, R. 1982 Natural convection in porous media. In Proc. Workshop on Natural Convection (ed. Yang, K. T. & Lloyd, J. R.), pp. 3645. University of Notre Dame, Indiana.
Whitaker, S. 1986 Flow in porous media: a theoretical derivation of Darcy's law. Transport in Porous Media 1, 325.
Xi, H.-W., Gunton, J. D. & Vinals, J. 1993 Spiral defect chaos in a model of Rayleigh-Benard convection. Phys. Rev. Lett. 71, 20302033.
Yen, Y. C. 1974 Effects of density inversion on free convective heat transfer in porous layer heated from below. Intl J. Heat. Mass Transfer 17, 13491356.
Zimmermann, W., Sesselberg, M. & Petruccione, F. 1993 Effect of disorder in pattern formation. Phys. Rev. E 71, 26992703. S13
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