Dynamic evolution of natural convection in a porous square cavity

doi:10.7523/j.ucas.2022.077

Abstract

Abstract: Natural convection in fluid-saturated porous cavities is a classical problem in the study of nonlinear hydrodynamics. Limited by the early computing capability, previous studies mainly focused on the dynamic mechanism of the spatiotemporal evolution of centrosymmetric convection modes, and cannot fully describe the dynamic behavior of real-world porous natural convection systems with all convection modes. Therefore, the available results cannot be directly applied to practical engineering problems. In this study, two-dimensional natural convection in a porous square cavity heated from below is numerically simulated with complete convection modes based on the Galerkin method and the collocation pseudo-spectral method, and the complete route for natural convection dynamically evolving from onset towards chaos is obtained for the first time. The modeling results reveal that along with Rayleigh number increasing from 4π²to 1200, the natural convection evolves across 21 different stages (including one steady convection stage, six single-frequency oscillation stages, nine quasiperiodic oscillation stages, two frequency-locked resonance stages, and three aperiodic chaotic oscillation stages). The spatial variation and temporal oscillation of convection modes are the fundamental reasons for the repeated alternation between different convection patterns. Different characteristics of single-frequency, quasiperiodic, frequency-locked and chaotic oscillations in the calculated heat flow represented by Nusselt number are systematically compared in terms of four graphical methods, viz., time series, power spectrum, phase portrait, and Lorenz mapping. The calculated average heat flow and primary oscillating frequency as functions of Rayleigh number can be fitted by simple analytical formulas (approximately obeying the scaling laws given by the classical boundary layer theory in a wide range of Rayleigh number), providing convenient calculation tools for practical engineering designs and applications under the corresponding conditions.

Key words: natural convection, dynamic evolution, porous media, convection mode, numerical simulation

CLC Number:

P314

ZHANG Lianping, WANG Shimin. Dynamic evolution of natural convection in a porous square cavity[J]. Journal of University of Chinese Academy of Sciences, DOI: 10.7523/j.ucas.2022.077.

References

[1] Horton C W, Rogers F T Jr. Convection currents in a porous medium[J]. Journal of Applied Physics, 1945, 16(6): 367-370. DOI:10.1063/1.1707601.
[2] Lapwood E R.Convection of a fluid in a porous medium[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1948, 44(4): 508-521. DOI:10.1017/S030500410002452X.
[3] 孔祥言. 高等渗流力学[M]. 合肥: 中国科学技术大学出版社, 1999.
[4] Phillips O M.Geological fluid dynamics: sub-surface flow and reactions[M]. Cambridge, UK: Cambridge University Press, 2009. DOI: DOI10.1017/cbo9780511807473.
[5] Nield D A, Bejan A.Convection in porous media[M]. 5th ed. Cham: Springer International Publishing, 2017. DOI:10.1007/978-3-319-49562-0.
[6] Neufeld J A, Hesse M A, Riaz A, et al.Convective dissolution of carbon dioxide in saline aquifers[J]. Geophysical Research Letters, 2010, 37(22): L22404. DOI:10.1029/2010GL044728.
[7] Akhbari D, Hesse M A.Causes of underpressure in natural CO₂ reservoirs and implications for geological storage[J]. Geology, 2017, 45(1): 47-50. DOI:10.1130/G38362.1.
[8] Huang X X, Zhu J L, Li J, et al.Fluid friction and heat transfer through a single rough fracture in granitic rock under confining pressure[J]. International Communications in Heat and Mass Transfer, 2016, 75: 78-85. DOI: 10.1016/j.icheatmasstransfer.2016.03.027.
[9] Wen B L, Corson L T, Chini G P.Structure and stability of steady porous medium convection at large Rayleigh number[J]. Journal of Fluid Mechanics, 2015, 772: 197-224. DOI:10.1017/jfm.2015.205.
[10] 孔祥言,卢德唐,徐献芝,等. 多孔介质中对流的研究[J]. 力学进展, 1996, 26(4): 510-520. DOI:10.6052/1000-0992-1996-4-J1996-048.
[11] Kimura S, Schubert G, Straus J M.Route to chaos in porous-medium thermal convection[J]. Journal of Fluid Mechanics, 1986, 166: 305-324. DOI:10.1017/S0022112086000162.
[12] Schubert G, Straus J M.Transitions in time-dependent thermal convection in fluid-saturated porous media[J]. Journal of Fluid Mechanics, 1982, 121: 301-313. DOI:10.1017/S0022112082001918.
[13] Kimura S, Schubert G, Straus J M.Instabilities of steady, periodic, and quasi-periodic modes of convection in porous media[J]. Journal of Heat Transfer, 1987, 109(2): 350-355. DOI:10.1115/1.3248087.
[14] Steen P H, Aidun C K.Time-periodic convection in porous media: transition mechanism[J]. Journal of Fluid Mechanics, 1988, 196: 263-290. DOI:10.1017/S0022112088002708.
[15] Graham M D, Steen P H.Strongly interacting travelling waves and quasiperiodic dynamics in porous medium convection[J]. Physica D: Nonlinear Phenomena, 1992, 54(4): 331-350. DOI:10.1016/0167-2789(92)90042-L.
[16] de la Torre Juárez M, Busse F H. Stability of two-dimensional convection in a fluid-saturated porous medium[J]. Journal of Fluid Mechanics, 1995, 292: 305-323. DOI:10.1017/S0022112095001534.
[17] Aidun C K, Steen P H.Transition to oscillatory convective heat transfer in a fluid-saturated porous medium[J]. Journal of Thermophysics and Heat Transfer, 1987, 1(3): 268-273. DOI:10.2514/3.38.
[18] Graham M D, Steen P H.Plume formation and resonant bifurcations in porous-media convection[J]. Journal of Fluid Mechanics, 1994, 272: 67-90. DOI:10.1017/S0022112094004386.
[19] Horne R N, Caltagirone J P.On the evolution of thermal disturbances during natural convection in a porous medium[J]. Journal of Fluid Mechanics, 1980, 100(2): 385-395. DOI:10.1017/S0022112080001218.
[20] Bories S.Natural convection in porous media[M]// Bear J and Corapcioglu M Y. Advances in transport phenomena in porous media. Dordrecht: Springer Netherlands, 1987: 77-141. DOI:10.1007/978-94-009-3625-6_4.
[21] Palm E, Weber J E, Kvernvold O.On steady convection in a porous medium[J]. Journal of Fluid Mechanics, 1972, 54(1): 153-161. DOI:10.1017/S002211207200059X.
[22] Orszag S A.Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations[J]. Studies in Applied Mathematics, 1971, 50(4): 293-327. DOI:10.1002/sapm1971504293.
[23] Ferziger J H, Perić M.Computational methods for fluid dynamics[M]. 3rd, rev. ed. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. DOI:10.1007/978-3-642-56026-2.
[24] Strogatz S.Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering[M]. 2nd ed. Boca Raton: CRC Press, 2018.
[25] Howard L N.Convection at high Rayleigh number[M]// Görtler H. Applied mechanics. Berlin, Heidelberg: Springer Berlin Heidelberg, 1966: 1109-1115. DOI:10.1007/978-3-662-29364-5_147.
[26] Buretta R J, Berman A S.Convective heat transfer in a liquid saturated porous layer[J]. Journal of Applied Mechanics, 1976, 43(2): 249-253. DOI:10.1115/1.3423818.
[27] Close D J, Peck M K.Experimental determination of the behaviour of wet porous beds in which natural convection occurs[J]. International Journal of Heat and Mass Transfer, 1986, 29(10): 1531-1541. DOI:10.1016/0017-9310(86)90068-2.
[28] Combarnous M A, Bories S A.Hydrothermal convection in saturated porous media[M]// Chow V T. Advances in hydroscience. Amsterdam: Elsevier, 1975: 231-307. DOI:10.1016/B978-0-12-021810-3.50008-4.
[29] Elder J W.Steady free convection in a porous medium heated from below[J]. Journal of Fluid Mechanics, 1967, 27(1): 29-48. DOI:10.1017/S0022112067000023.
[30] Kaneko T, Mohtadi M F, Aziz K.An experimental study of natural convection in inclined porous media[J]. International Journal of Heat and Mass Transfer, 1974, 17(4): 485-496. DOI:10.1016/0017-9310(74)90025-8.
[31] Kazmierczak M, Muley A.Steady and transient natural convection experiments in a horizontal porous layer: the effects of a thin top fluid layer and oscillating bottom wall temperature[J]. International Journal of Heat and Fluid Flow, 1994, 15(1): 30-41. DOI:10.1016/0142-727X(94)90028-0.
[32] Lister C R B. An explanation for the multivalued heat transport found experimentally for convection in a porous medium[J]. Journal of Fluid Mechanics, 1990, 214: 287-320. DOI:10.1017/S0022112090000143.
[33] Schneider K J.Investigation of the influence of free thermal convection on heat transfer through granular material[M]// Proceedings of the XIth International Congress of Refrigeration. Progress in refrigeration science and technology. Amsterdam: Elsevier, 1965: 247-254. DOI:10.1016/b978-1-4831-9857-6.50053-4.
[34] Seki N, Fukusako S, Ariake Y.Experimental study of free convective heat transfer in a liquid-saturated porous bed at high Rayleigh number[J]. Wӓrme und Stoffübertragung, 1980, 13(1): 61-71. DOI:10.1007/BF00997634.
[35] Yen Y C.Effects of density inversion on free convective heat transfer in porous layer heated from below[J]. International Journal of Heat and Mass Transfer, 1974, 17(11): 1349-1356. DOI:10.1016/0017-9310(74)90136-7.
[36] 丁军锋, 王世民. 增强型地热系统的多区域多物理场耦合三维数值模拟[J]. 中国科学院大学学报, 2019, 36(5): 694-701. DOI:10.7523/j.issn.2095-6134.2019.05.015.