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From Rayleigh–Bénard convection to porous-media convection: how porosity affects heat transfer and flow structure

  • Shuang Liu (a1) (a2), Linfeng Jiang (a1), Kai Leong Chong (a3), Xiaojue Zhu (a4), Zhen-Hua Wan (a5), Roberto Verzicco (a3) (a6) (a7), Richard J. A. M. Stevens (a3), Detlef Lohse (a3) (a8) and Chao Sun (a1) (a2)...

Abstract

We perform a numerical study of the heat transfer and flow structure of Rayleigh–Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $\unicode[STIX]{x1D719}$ in the flow properties during the transition process from the traditional RB convection with $\unicode[STIX]{x1D719}=1$ (so no obstacles included) to Darcy-type porous-media convection with $\unicode[STIX]{x1D719}$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for Rayleigh number $Ra$ from $10^{5}$ to $10^{10}$ and varying porosities $\unicode[STIX]{x1D719}$ , at a fixed Prandtl number $Pr=4.3$ , and we restrict ourselves to the two-dimensional case. For fixed $Ra$ , the Nusselt number $Nu$ is found to vary non-monotonically as a function of $\unicode[STIX]{x1D719}$ ; namely, with decreasing $\unicode[STIX]{x1D719}$ , it first increases, before it decreases for $\unicode[STIX]{x1D719}$ approaching 0. The non-monotonic behaviour of $Nu(\unicode[STIX]{x1D719})$ originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed  $\unicode[STIX]{x1D719}$ , depending on $Ra$ , two different heat transfer regimes are identified, with different effective power-law behaviours of $Nu$ versus $Ra$ , namely a steep one for low $Ra$ when viscosity dominates, and the standard classical one for large $Ra$ . The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.

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Corresponding author

Email address for correspondence: chaosun@tsinghua.edu.cn

References

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Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.
Amooie, M. A., Soltanian, M. R. & Moortgat, J. 2018 Solutal convection in porous media: comparison between boundary conditions of constant concentration and constant flux. Phys. Rev. E 98 (3), 033118.
Araújo, A. D., Bastos, W. B., Andrade, J. S. Jr. & Herrmann, H. J. 2006 Distribution of local fluxes in diluted porous media. Phys. Rev. E 74 (1), 010401.
Ardekani, M. N., Abouali, O., Picano, F. & Brandt, L. 2018a Heat transfer in laminar Couette flow laden with rigid spherical particles. J. Fluid Mech. 834, 308334.
Ardekani, M. N., Al Asmar, L., Picano, F. & Brandt, L. 2018b Numerical study of heat transfer in laminar and turbulent pipe flow with finite-size spherical particles. Intl J. Heat Fluid Flow 71, 189199.
Ataei-Dadavi, I., Chakkingal, M., Kenjeres, S., Kleijn, C. R. & Tummers, M. J. 2019 Flow and heat transfer measurements in natural convection in coarse-grained porous media. Intl J. Heat Mass Transfer 130, 575584.
Bao, Y., Chen, J., Liu, B.-F., She, Z.-S., Zhang, J. & Zhou, Q. 2015 Enhanced heat transport in partitioned thermal convection. J. Fluid Mech. 784, R5.
Breugem, W.-P. 2012 A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows. J. Comput. Phys. 231 (13), 44694498.
Chakkingal, M., Kenjereš, S., Ataei-Dadavi, I., Tummers, M. J. & Kleijn, C. R. 2019 Numerical analysis of natural convection with conjugate heat transfer in coarse-grained porous media. Intl J. Heat Fluid Flow 77, 4860.
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (7), 58.
Chong, K. L., Huang, S.-D., Kaczorowski, M. & Xia, K.-Q. 2015 Condensation of coherent structures in turbulent flows. Phys. Rev. Lett. 115 (26), 264503.
Chong, K. L., Wagner, S., Kaczorowski, M., Shishkina, O. & Xia, K.-Q. 2018 Effect of Prandtl number on heat transport enhancement in Rayleigh–Bénard convection under geometrical confinement. Phys. Rev. F 3 (1), 013501.
Chong, K. L., Yang, Y.-T., Huang, S.-D., Zhong, J.-Q., Stevens, R. J. A. M., Verzicco, R., Lohse, D. & Xia, K.-Q. 2017 Confined Rayleigh–Bénard, rotating Rayleigh–Bénard, and double diffusive convection: a unifying view on turbulent transport enhancement through coherent structure manipulation. Phys. Rev. Lett. 119 (6), 064501.
Cinar, Y. & Riaz, A. 2014 Carbon dioxide sequestration in saline formations. Part 2. Review of multiphase flow modeling. J. Petrol. Sci. Engng 124, 381398.
Cinar, Y., Riaz, A. & Tchelepi, H. A. 2009 Experimental study of CO2 injection into saline formations. SPE J. 14, 588594.
De Paoli, M., Zonta, F. & Soldati, A. 2016 Influence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. Phys. Fluids 28 (5), 056601.
Emami-Meybodi, H. & Hassanzadeh, H. 2015 Two-phase convective mixing under a buoyant plume of CO2 in deep saline aquifers. Adv. Water Resour. 76, 5571.
Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd-Yusof, J. 2000 Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161 (1), 3560.
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86 (15), 33163319.
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16 (12), 44624472.
Hassanzadeh, H., Pooladidarvish, M. & Keith, D. W. 2007 Scaling behavior of convective mixing, with application to geological storage of CO2. AIChE J. 53 (5), 11211131.
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2012 Ultimate regime of high Rayleigh number convection in a porous medium. Phys. Rev. Lett. 108, 224503.
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2014 High Rayleigh number convection in a three-dimensional porous medium. J. Fluid Mech. 748, 879895.
Huang, S.-D., Kaczorowski, M., Ni, R. & Xia, K.-Q. 2013 Confinement-induced heat-transport enhancement in turbulent thermal convection. Phys. Rev. Lett. 111 (10), 104501.
Huang, Y.-X. & Zhou, Q. 2013 Counter-gradient heat transport in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 737, R3.
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255272.
Jiang, H.-C., Zhu, X.-J., Mathai, V., Verzicco, R., Lohse, D. & Sun, C. 2018 Controlling heat transport and flow structures in thermal turbulence using ratchet surfaces. Phys. Rev. Lett. 120 (4), 044501.
Jiang, L.-F., Sun, C. & Calzavarini, E. 2019 Robustness of heat transfer in confined inclined convection at high Prandtl number. Phys. Rev. E 99 (1), 013108.
Joseph, D. D., Nield, D. A. & Papanicolaou, G. 1982 Nonlinear equation governing flow in a saturated porous medium. Adv. Water Resour. 18 (4), 10491052.
Keene, D. J. & Goldstein, R. J. 2015 Thermal convection in porous media at high Rayleigh numbers. J. Heat Transfer. 137 (3), 034503.
Kempe, T. & Fröhlich, J. 2012 An improved immersed boundary method with direct forcing for the simulation of particle laden flows. J. Comput. Phys. 231 (9), 36633684.
Lakkaraju, R., Stevens, R. J. A. M., Oresta, P., Verzicco, R., Lohse, D. & Prosperetti, A. 2013 Heat transport in bubbling turbulent convection. Proc. Natl Acad. Sci. USA 110 (23), 92379242.
Landman, A. J. & Schotting, R. J. 2007 Heat and brine transport in porous media: the Oberbeck–Boussinesq approximation revisited. Trans. Porous Med. 70 (3), 355373.
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508521.
Lim, Z.-L., Chong, K. L., Ding, G.-Y. & Xia, K.-Q. 2019 Quasistatic magnetoconvection: heat transport enhancement and boundary layer crossing. J. Fluid Mech. 870, 519542.
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.
Nithiarasu, P., Seetharamu, K. N. & Sundararajan, T. 1997 Natural convective heat transfer in a fluid saturated variable porosity medium. Intl J. Heat Mass Transfer 40 (16), 39553967.
Orr, F. M. 2009 Onshore geologic storage of CO2. Science 325 (5948), 16561658.
Otero, J., Dontcheva, L. A., Johnston, H., Worthing, R. A., Kurganov, A., Petrova, G. & Doering, C. R. 2004 High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263281.
van der Poel, E. P., Ostilla-Mónico, R., Donners, J. & Verzicco, R. 2015 A pencil distributed finite difference code for strongly turbulent wall-bounded flows. Comput. Fluids 116, 1016.
van der Poel, E. P., Stevens, R. J. A. M. & Lohse, D. 2013 Comparison between two and three dimensional Rayleigh–Bénard convection. J. Fluid Mech. 736, 177194.
Riaz, A. & Cinar, Y. 2014 Carbon dioxide sequestration in saline formations. Part I. Review of the modeling of solubility trapping. J. Petrol. Sci. Engng 124, 367380.
Sardina, G., Brandt, L., Boffetta, G. & Mazzino, A. 2018 Buoyancy-driven flow through a bed of solid particles produces a new form of Rayleigh–Taylor turbulence. Phys. Rev. Lett. 121 (22), 224501.
Shishkina, O. & Horn, S. 2016 Thermal convection in inclined cylindrical containers. J. Fluid Mech. 790, R3.
Shishkina, O., Stevens, R. J. A. M., Grossmann, S. & Lohse, D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys. 12 (7), 075022.
Shishkina, O. & Wagner, C. 2011 Modelling the influence of wall roughness on heat transfer in thermal convection. J. Fluid Mech. 686, 568582.
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42 (6), 36503653.
Soltanian, M. R., Amooie, M. A., Dai, Z.-X., Cole, D. & Moortgat, J. 2016 Critical dynamics of gravito-convective mixing in geological carbon sequestration. Sci. Rep. 6, 35921.
Spandan, V., Lohse, D., de Tullio, M. D. & Verzicco, R. 2018 A fast moving least squares approximation with adaptive Lagrangian mesh refinement for large scale immersed boundary simulations. J. Comput. Phys. 375, 228239.
Spandan, V., Meschini, V., Ostilla-Mónico, R., Lohse, D., Querzoli, G., de Tullio, M. D. & Verzicco, R. 2017 A parallel interaction potential approach coupled with the immersed boundary method for fully resolved simulations of deformable interfaces and membranes. J. Comput. Phys. 348, 567590.
Stevens, R. J. A. M., Lohse, D. & Verzicco, R. 2014 Sidewall effects in Rayleigh–Bénard convection. J. Fluid Mech. 741, 127.
Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495507.
Stevens, R. J. A. M., Zhong, J.-Q., Clercx, H. J. H., Ahlers, G. & Lohse, D. 2009 Transitions between turbulent states in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 103 (2), 024503.
Sugiyama, K., Ni, R., Stevens, R. J. A. M., Chan, T. S., Zhou, S.-Q., Xi, H.-D., Sun, C., Grossmann, S., Xia, K.-Q. & Lohse, D. 2010 Flow reversals in thermally driven turbulence. Phys. Rev. Lett. 105 (3), 034503.
de Tullio, M. D. & Pascazio, G. 2016 A moving-least-squares immersed boundary method for simulating the fluid–structure interaction of elastic bodies with arbitrary thickness. J. Comput. Phys. 325, 201225.
Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209 (2), 448476.
Vanella, M. & Balaras, E. 2009 Short note: a moving-least-squares reconstruction for embedded-boundary formulations. J. Comput. Phys. 228 (18), 66176628.
Verzicco, R. 2002 Sidewall finite-conductivity effects in confined turbulent thermal convection. J. Fluid Mech. 473, 201210.
Verzicco, R. 2004 Effects of nonperfect thermal sources in turbulent thermal convection. Phys. Fluids 16 (6), 19651979.
Wagner, S. & Shishkina, O. 2015 Heat flux enhancement by regular surface roughness in turbulent thermal convection. J. Fluid Mech. 763, 109135.
Wang, Q., Wan, Z.-H., Yan, R. & Sun, D.-J. 2018 Multiple states and heat transfer in two-dimensional tilted convection with large aspect ratios. Phys. Rev. Fluids 3 (11), 113503.
Wang, Z.-Q., Mathai, V. & Sun, C. 2019 Self-sustained biphasic catalytic particle turbulence. Nat. Commun. 10 (1), 3333.
Wen, B.-L., Corson, L. T. & Chini, G. P. 2015 Structure and stability of steady porous medium convection at large Rayleigh number. J. Fluid Mech. 772, 197224.
Wooding, R. A. 1957 Steady state free thermal convection of liquid in a saturated permeable medium. J. Fluid Mech. 2 (3), 273285.
Xia, K.-Q. 2013 Current trends and future directions in turbulent thermal convection. Theor. Appl. Mech. Lett. 3 (5), 052001.
Yang, Y.-T., van der Poel, E. P., Ostilla-Mónico, R., Sun, C., Verzicco, R., Grossmann, S. & Lohse, D. 2015 Salinity transfer in bounded double diffusive convection. J. Fluid Mech. 768, 476491.
Yang, Y.-T., Verzicco, R. & Lohse, D. 2016 Scaling laws and flow structures of double diffusive convection in the finger regime. J. Fluid Mech. 802, 667689.
Zhang, Y., Zhou, Q. & Sun, C. 2017 Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 814, 165184.
Zhong, J.-Q., Stevens, R. J. A. M., Clercx, H. J. H., Verzicco, R., Lohse, D. & Ahlers, G. 2009 Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 102 (4), 044502.
Zhu, X.-J., Stevens, R. J. A. M., Shishkina, O., Verzicco, R. & Lohse, D. 2019 NuRa 1/2 scaling enabled by multiscale wall roughness in Rayleigh–Bénard turbulence. J. Fluid Mech. 869, R4.
Zhu, X.-J., Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2017 Roughness-facilitated local 1/2 scaling does not imply the onset of the ultimate regime of thermal convection. Phys. Rev. Lett. 119 (15), 154501.
Zwirner, L. & Shishkina, O. 2018 Confined inclined thermal convection in low-Prandtl-number fluids. J. Fluid Mech. 850, 9841008.
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From Rayleigh–Bénard convection to porous-media convection: how porosity affects heat transfer and flow structure

  • Shuang Liu (a1) (a2), Linfeng Jiang (a1), Kai Leong Chong (a3), Xiaojue Zhu (a4), Zhen-Hua Wan (a5), Roberto Verzicco (a3) (a6) (a7), Richard J. A. M. Stevens (a3), Detlef Lohse (a3) (a8) and Chao Sun (a1) (a2)...

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