流体饱和孔隙介质腔体内的自然对流是非线性流体动力学研究的经典问题。受早期计算条件限制，前人研究主要侧重于中心对称对流模态的时空演化动力学机制，不能完整刻画实际孔隙自然对流系统中包含全部对流模态的动力学行为，因而无法直接应用于实际工程。基于伽辽金法和配点伪谱法数值模拟底部加热的孔隙介质方腔内模态完备的二维自然对流，首次得到自然对流从发生到走向混沌的完整动力学演化路径。模拟结果揭示，随着Ra从4π2单调增加到1 200，自然对流演化历经21个不同阶段(包括1个稳态对流阶段、6个单频振荡阶段、9个准周期振荡阶段、2个锁频共振阶段和3个非周期混沌振荡阶段)。对流模态的空间变化与时间振荡是对流形态反复发生交替的根本原因。采用时间序列、功率谱、相图和洛伦兹映射4种作图方法，系统对比Nu表征的计算热流作单频、准周期、锁频和混沌振荡的不同特征。计算平均热流和振荡主频随Ra的变化可由简单的解析公式拟合(在很宽的Ra区间内近似服从经典边界层标度律)，为相应条件下实际工程设计和应用提供了便捷的计算工具。
Natural convection in fluid-saturated porous cavities is a classical problem in the study of nonlinear hydrodynamics. Limited by the early computing capabilities, previous studies mainly focused on the dynamic mechanism of the spatiotemporal evolution of centrosymmetric convection modes, and could not fully describe the dynamic behavior of real-world porous natural convection systems with all convection modes. Therefore, the available results could not 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 Ra increasing from 4π2 to 1 200, 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 Nu 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 Ra can be fitted by simple analytical formulas, approximately obeying the scaling laws given by the classical boundary layer theory in a wide range of Ra values. This provides convenient calculation tools for practical engineering designs and applications under the corresponding conditions.
[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:10.1017/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, 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, Peric'M. Computational methods for fluid dynamics[M]. 3rd 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]. Wärme 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.
