Hostname: page-component-cc8bf7c57-n7pht Total loading time: 0 Render date: 2024-12-11T00:31:37.911Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  18 May 2020

Shuang Liu
Affiliation:
Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, International Joint Laboratory on Low Carbon Clean Energy Innovation, Department of Energy and Power Engineering, Tsinghua University, Beijing100084, China Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing100084, China
Linfeng Jiang
Affiliation:
Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, International Joint Laboratory on Low Carbon Clean Energy Innovation, Department of Energy and Power Engineering, Tsinghua University, Beijing100084, China
Kai Leong Chong
Affiliation:
Physics of Fluids Group and Max Planck Center Twente, University of Twente, PO Box 217, 7500AEEnschede, The Netherlands
Xiaojue Zhu
Affiliation:
Center of Mathematical Sciences and Applications, and School of Engineering and Applied Sciences, Harvard University, Cambridge, MA02138, USA
Zhen-Hua Wan
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui230027, China
Roberto Verzicco
Affiliation:
Physics of Fluids Group and Max Planck Center Twente, University of Twente, PO Box 217, 7500AEEnschede, The Netherlands Dipartimento di Ingegneria Industriale, University of Rome ‘Tor Vergata’, Via del Politecnico 1, 00133Roma, Italy Gran Sasso Science Institute, Viale F. Crispi 7, 67100L’Aquila, Italy
Richard J. A. M. Stevens
Affiliation:
Physics of Fluids Group and Max Planck Center Twente, University of Twente, PO Box 217, 7500AEEnschede, The Netherlands
Detlef Lohse
Affiliation:
Physics of Fluids Group and Max Planck Center Twente, University of Twente, PO Box 217, 7500AEEnschede, The Netherlands Max Planck Institute for Dynamics and Self-Organization, 37077Göttingen, Germany
Chao Sun*
Affiliation:
Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, International Joint Laboratory on Low Carbon Clean Energy Innovation, Department of Energy and Power Engineering, Tsinghua University, Beijing100084, China Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing100084, China
*
Email address for correspondence: chaosun@tsinghua.edu.cn

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.

Type
JFM Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.CrossRefGoogle Scholar
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.Google Scholar
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.Google ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (7), 58.Google ScholarPubMed
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.CrossRefGoogle ScholarPubMed
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.Google Scholar
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.CrossRefGoogle ScholarPubMed
Cinar, Y. & Riaz, A. 2014 Carbon dioxide sequestration in saline formations. Part 2. Review of multiphase flow modeling. J. Petrol. Sci. Engng 124, 381398.CrossRefGoogle Scholar
Cinar, Y., Riaz, A. & Tchelepi, H. A. 2009 Experimental study of CO2 injection into saline formations. SPE J. 14, 588594.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86 (15), 33163319.CrossRefGoogle ScholarPubMed
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16 (12), 44624472.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Huang, Y.-X. & Zhou, Q. 2013 Counter-gradient heat transport in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 737, R3.CrossRefGoogle Scholar
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255272.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
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.Google ScholarPubMed
Joseph, D. D., Nield, D. A. & Papanicolaou, G. 1982 Nonlinear equation governing flow in a saturated porous medium. Adv. Water Resour. 18 (4), 10491052.Google Scholar
Keene, D. J. & Goldstein, R. J. 2015 Thermal convection in porous media at high Rayleigh numbers. J. Heat Transfer. 137 (3), 034503.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
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.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508521.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.CrossRefGoogle Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.Google Scholar
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.CrossRefGoogle Scholar
Orr, F. M. 2009 Onshore geologic storage of CO2. Science 325 (5948), 16561658.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Shishkina, O. & Horn, S. 2016 Thermal convection in inclined cylindrical containers. J. Fluid Mech. 790, R3.CrossRefGoogle Scholar
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.Google Scholar
Shishkina, O. & Wagner, C. 2011 Modelling the influence of wall roughness on heat transfer in thermal convection. J. Fluid Mech. 686, 568582.CrossRefGoogle Scholar
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42 (6), 36503653.CrossRefGoogle ScholarPubMed
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.Google ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Stevens, R. J. A. M., Lohse, D. & Verzicco, R. 2014 Sidewall effects in Rayleigh–Bénard convection. J. Fluid Mech. 741, 127.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209 (2), 448476.CrossRefGoogle Scholar
Vanella, M. & Balaras, E. 2009 Short note: a moving-least-squares reconstruction for embedded-boundary formulations. J. Comput. Phys. 228 (18), 66176628.CrossRefGoogle Scholar
Verzicco, R. 2002 Sidewall finite-conductivity effects in confined turbulent thermal convection. J. Fluid Mech. 473, 201210.CrossRefGoogle Scholar
Verzicco, R. 2004 Effects of nonperfect thermal sources in turbulent thermal convection. Phys. Fluids 16 (6), 19651979.CrossRefGoogle Scholar
Wagner, S. & Shishkina, O. 2015 Heat flux enhancement by regular surface roughness in turbulent thermal convection. J. Fluid Mech. 763, 109135.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Wang, Z.-Q., Mathai, V. & Sun, C. 2019 Self-sustained biphasic catalytic particle turbulence. Nat. Commun. 10 (1), 3333.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Wooding, R. A. 1957 Steady state free thermal convection of liquid in a saturated permeable medium. J. Fluid Mech. 2 (3), 273285.CrossRefGoogle Scholar
Xia, K.-Q. 2013 Current trends and future directions in turbulent thermal convection. Theor. Appl. Mech. Lett. 3 (5), 052001.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Zwirner, L. & Shishkina, O. 2018 Confined inclined thermal convection in low-Prandtl-number fluids. J. Fluid Mech. 850, 9841008.CrossRefGoogle Scholar