热浮力羽流的生成与演化: 从个体运动到群体行为

热对流普遍存在于自然界和工程应用中，Rayleigh-Bénard 系统是研究此现
象的经典模型。在Rayleigh-Bénard 系统中，从边界生成的羽流（具有一定浮力
的流体微团）在特定条件下会自发地组织成大尺度流动结构，其对促进物质和
能量的传递及交换起到了关键作用。因此，理解该大尺度自组织结构的形成机
制，对于调控对流输运和换热效率具有重要意义。本文通过将羽流视为基本的
自驱动单元，采用实验和数值模拟相结合的手段，系统地研究了羽流从个体到
群体的生成演化规律，并构建了活性羽流模型。本文的具体研究结果如下：

首先，本文对单羽流的特性进行了研究，考察了单羽流的生成过程及其与
边界层发展演化的关联，并重点测量了瑞利数和普朗特数对单羽流运动特征的
影响。这些结果为理解羽流间的相互作用及活性羽流模型的构建提供了基本认
知。随后，本文研究了双羽流间的相互作用，分析了羽流相互作用势的大小及
其空间分布特征，并通过考察羽流间隔以及羽流强度对相互作用过程的影响，
明确了双羽流间相互作用的有效范围和强度。在此基础上，本文进一步地研究
了群体羽流的自组织行为，分析了羽流自组织过程中流态的变化特点，总结了
典型的发展阶段及其特征。通过考察羽流间隔、羽流强度以及羽流数量对流态
及系统关联尺度的影响，本文明确了群体羽流从无序到有序的自组织临界点。

最后，本文借鉴了活性物质系统中的代理模型方法，通过将单羽流的生成
演化规律赋予到代理个体中，并加上羽流间的相互作用特征，建立了活性羽流
模型。该模型基本复现了实验和数值模拟中观测到的自组织流动结构，证明了
模型的合理性与有效性。利用活性羽流模型，本文进一步研究了相互作用强度
对自组织临界点的影响，发现羽流间相互作用强度是羽流自组织行为的关键动
力，且自组织结构具有一定的鲁棒性。

综上，本文以热对流系统中的基本流动单元——羽流为研究对象，通过结
合实验测量、数值模拟以及模型仿真的方法，对羽流从个体到群体的动力学行
为进行了系统地研究，从而揭示了羽流通过短程相互作用自组织形成大尺度流
动结构的动力学机制。相关结果不仅可以用于热对流系统中大尺度流动结构的
调控，还为研究其它流体系统中的自组织动力学结构提供了新的思路。

Thermal convection is a prevalent phenomenon in both nature and engineering applications. The Rayleigh-Bénard system serves as a classic model for investigating these phenomena. Within the Rayleigh-Bénard system, plumes, which are fluid parcels with a distinct buoyancy, spontaneously organize into large-scale flow structures under specific conditions, playing a pivotal role in facilitating material and energy exchange and transfer. Consequently, understanding the mechanisms governing the formation of these large-scale self-organized structures holds paramount significance for regulating convection-driven transport and heat exchange efficiency. In this paper, by treating plumes as fundamental self-driven units, we systematically investigate the generation and evolution of plumes from individual to collective behavior through a combination of experimental and numerical simulation methods, and construct an active plume model. The specific findings of this study are detailed below.

To commence, this study delves into the characteristics of the fundamental unit of
single plumes. It examines the process of single plume generation and its association with the boundary layer. Significantly, the influence of Rayleigh number and Prandtl number on the motion features of single plumes is meticulously measured. These results provide a fundamental understanding that underpins the analysis of plume interactions and the construction of the active plume model. Subsequently, the interactions between pairs of plumes are investigated. The study analyzes the magnitude and spatial distribution characteristics of plume interaction potentials. By examining the impact of plume spacing and plume strength on the interaction process, the range and intensity of interactions between two plumes are defined. Building upon these foundations, this research further explores the self-organizing behavior of plume ensembles, analyzing the changing characteristics of flow states during the process. It identifies and characterizes the typical stages of development. By investigating the effects of plume spacing, plume strength, and plume number on the flow states and the system’s correlation scales, the critical point of self-organization of plume ensembles from disorder to order is precisely defined.

Finally, the study draws inspiration from classical agent models in active matter
systems. By endowing agent individuals with the generation and evolution laws of single plumes and incorporating plume interactions, an active plume model is established.
The model successfully replicates the observed self-organized flow structures in both
experimental and numerical scenarios, affirming its validity and effectiveness. Utilizing
this active plume model, the research delves into the impact of interaction strength on the critical point of self-organization and discovers the robustness of plume self-organization dynamics.

In summary, this research focuses on plumes, the basic flow units in thermal convection
systems. By combining experimental measurements, numerical simulations, and
model simulations, it offers a comprehensive investigation of the dynamic behavior of
plumes from the individual to the collective level. This endeavor unveils the dynamic
mechanisms through which plumes, by virtue of short-range interactions, self-organize
into large-scale flow structures. The implications of these findings extend beyond the control of large-scale flow structures in thermal convection systems, providing novel insights into self-organizing dynamics within other fluid systems.

[1] GRIFFITHS R. Dynamics of mantle thermals with constant buoyancy or anomalousinternal heating[J]. Earth and Planetary Science Letters, 1986, 78(4):435-446.
[2] GRIFFITHS R W, CAMPBELL I H. Stirring and structure in mantle startingplumes[J]. Earth and Planetary Science Letters, 1990, 99(1-2):66-78.
[3] COULLIETTE D, LOPER D. Experimental, numerical and analytical models ofmantle starting plumes[J]. Physics of the Earth and Planetary Interiors, 1995, 92(3-4):143-167.
[4] JELLINEK A M,MANGA M. Links between long-lived hot spots, mantle plumes,𝐷′′, and plate tectonics[J]. Reviews of Geophysics, 2004, 42(3).
[5] CARDIN P, OLSON P. Chaotic thermal convection in a rapidly rotating sphericalshell: Consequences for flowin the outer core[J]. Physics of the Earth and PlanetaryInteriors, 1994, 82(3-4):0-259.
[6] HELFRICH K R. Thermals with background rotation and stratification[J]. Journalof Fluid Mechanics, 1994, 259:265-280.
[7] HORNER-DEVINE A R, FONG D A, MONISMITH S G, et al. Laboratoryexperiments simulating a coastal river inflow[J]. Journal of Fluid Mechanics,2006, 555:203.
[8] COLES V J, BROOKSMT, HOPKINS J, et al. The pathways and properties of theamazon river plume in the tropical north atlantic ocean[J]. Journal of GeophysicalResearch: Oceans, 2013, 118(12):6894-6913.
[9] HEWITT I J. Subglacial plumes[J]. Annual Review of Fluid Mechanics, 2020,52:145-169.
[10] MCCARTHY J J, CANZIANI O, LEARY N, et al. Intergovernmental panel onclimate change, 2001[J]. Working Group II. Climate change, 2001.
[11] SLAWSON P, CSANADY G. The effect of atmospheric conditions on plume rise[J]. Journal of Fluid Mechanics, 1971, 47(1):33-49.
[12] COMMONSW. File:nasa depiction of earth global atmospheric circulation.jpg—wikimedia commons, the free media repository[EB/OL]. 2022. https://commons.wikimedia.org/w/index.php?title=File:NASA_depiction_of_earth_global_atmospheric_circulation.jpg&oldid=709572113.
[13] COMMONS W. File:oceanic spreading.png — wikimedia commons, the freemedia repository[EB/OL]. 2019. https://commons.wikimedia.org/w/index.php?title=File:Oceanic_spreading.png&oldid=341648199.
[14] AHLERS G, GROSSMANN S, LOHSE D. Heat transfer and large scale dynamicsin turbulent rayleigh-bénard convection[J]. Reviews of Modern Physics, 2009, 81(2):503.
[15] XIA K Q. Current trends and future directions in turbulent thermal convection[J].Theoretical and Applied Mechanics Letters, 2013, 3(5):052001.
[16] LI X M, HE J D, TIAN Y, et al. Effects of prandtl number in quasi-two-dimensionalrayleigh–bénard convection[J]. Journal of Fluid Mechanics, 2021, 915.
[17] KAWAGUCHI K, KAGEYAMA R, SANO M. Topological defects control collectivedynamics in neural progenitor cell cultures[J]. Nature, 2017, 545(7654):327-331.
[18] GRÉGOIRE G, CHATÉ H. Onset of collective and cohesive motion[J]. PhysicalReview Letters, 2004, 92(2):025702.
[19] GINOT F, THEURKAUFF I, LEVIS D, et al. Nonequilibrium equation of state insuspensions of active colloids[J]. Physical Review X, 2015, 5(1):011004.
[20] BIALKÉ J, SPECK T, LÖWEN H. Active colloidal suspensions: Clustering andphase behavior[J]. Journal of Non-Crystalline Solids, 2015, 407:367-375.
[21] ZHANG H P, BE’ER A, FLORIN E L, et al. Collective motion and densityfluctuations in bacterial colonies[J]. Proceedings of the National Academy ofSciences, 2010, 107(31):13626-13630.
[22] BECHINGER C, DI LEONARDOR,LÖWENH, et al. Active particles in complexand crowded environments[J]. Reviews of Modern Physics, 2016, 88(4):045006.
[23] LIU S, SHANKAR S, MARCHETTIMC, et al. Viscoelastic control of spatiotemporalorder in bacterial active matter[J]. Nature, 2021, 590(7844):80-84.
[24] CHEN X, DONG X, BE’ER A, et al. Scale-invariant correlations in dynamicbacterial clusters[J]. Physical Review Letters, 2012, 108(14):148101.
[25] NISHIGUCHI D, ARANSON I S, SNEZHKO A, et al. Engineering bacterialvortex lattice via direct laser lithography[J]. Nature Communications, 2018, 9(1):1-8.
[26] SCHALLERV,WEBERC,SEMMRICHC, et al. Polar patterns of driven filaments[J]. Nature, 2010, 467(7311):73-77.
[27] OPATHALAGE A,NORTONMM, JUNIPERMP, et al. Self-organized dynamicsand the transition to turbulence of confined active nematics[J]. Proceedings of theNational Academy of Sciences, 2019, 116(11):4788-4797.
[28] DELSUC F. Army ants trapped by their evolutionary history[J]. PLoS Biology,2003, 1(2):e37.
[29] DIBENSKI D. Auklet flock, shumagins[EB/OL]. 2006. https://commons.wikimedia.org/w/index.php?title=File:Oceanic_spreading.png&oldid=341648199.
[30] HAAS R B. Flamingos, yucatan peninsula[EB/OL]. 2007. https://www.nationalgeographic.com/photo-of-the-day/photo/flamingos-gulf-mexico.
[31] HUGHES R. Barracuda tornado[EB/OL]. 2007. https://www.flickr.com/photos/robinhughes/404457553/in/photolist-BJX96-CJZAK.
[32] CHANDRAGIRI S,DOOSTMOHAMMADIA,YEOMANSJ M, et al. Flowstatesand transitions of an active nematic in a three-dimensional channel[J]. PhysicalReview Letters, 2020, 125(14):148002.
[33] ZHANG B, YUAN H, SOKOLOV A, et al. Polar state reversal in active fluids[J].Nature Physics, 2022, 18(2):154-159.
[34] WORKAMP M, RAMIREZ G, DANIELS K E, et al. Symmetry-reversals in chiralactive matter[J]. Soft Matter, 2018, 14(27):5572-5580.
[35] BAIN N, BARTOLO D. Dynamic response and hydrodynamics of polarizedcrowds[J]. Science, 2019, 363(6422):46-49.
[36] OUELLETTE N T. Flowing crowds[J]. Science, 2019, 363(6422):27-28.
[37] REYNOLDS CW. Flocks, herds and schools: A distributed behavioral model[C]//Proceedings of the 14th annual conference on Computer graphics and interactivetechniques. [S.l.: s.n.], 1987: 25-34.
[38] VICSEK T, ZAFEIRIS A. Collective motion[J]. Physics Reports, 2012, 517(3-4):71-140.
[39] TONER J, TU Y. Long-range order in a two-dimensional dynamical xy model:how birds fly together[J]. Physical Review Letters, 1995, 75(23):4326.
[40] PROST J, JÜLICHER F, JOANNY J F. Active gel physics[J]. Nature Physics,2015, 11(2):111-117.
[41] BÉNARDH. Les tourbillons cellulaires dans une nappe liquide[J]. Revue Généraledes Sciences Pures et Appliquées, 1900, 11:1261-1271.
[42] RAYLEIGH L. Lix. on convection currents in a horizontal layer of fluid, when thehigher temperature is on the under side[J]. The London, Edinburgh, and DublinPhilosophical Magazine and Journal of Science, 1916, 32(192):529-546.
[43] KRISHNAMURTI R, HOWARD L N. Large-scale flow generation in turbulentconvection[J]. Proceedings of the National Academy of Sciences, 1981, 78(4):1981-1985.
[44] XIA K Q. Current trends and future directions in turbulent thermal convection[J].Theoretical and Applied Mechanics Letters, 2013, 3(5):052001.
[45] BÉNARDH. Les tourbillons cellulaires dans une nappe liquide.-méthodes optiquesd’observation et d’enregistrement[J]. Journal de Physique Théorique et Appliquée,1901, 10(1):254-266.
[46] SHANG X D, QIU X L, TONG P, et al. Measured local heat transport in turbulentrayleigh-bénard convection[J]. Physical Review Letters, 2003, 90(7):074501.
[47] VASILIEV A, SUKHANOVSKII A. Turbulent convection in a cube with mixedthermal boundary conditions: lowrayleigh number regime[J]. International Journalof Heat and Mass Transfer, 2021, 174:121290.
[48] GOLUSKIN D, DOERING C R. Bounds for convection between rough boundaries[J]. Journal of Fluid Mechanics, 2016, 804:370-386.
[49] URBAN P, HANZELKA P, KRÁLIK T, et al. Thermal waves and heat transferefficiency enhancement in harmonically modulated turbulent thermal convection[J]. Physical Review Letters, 2022, 128(13):134502.
[50] AHLERS G, BODENSCHATZ E, HARTMANN R, et al. Aspect ratio dependenceof heat transfer in a cylindrical rayleigh-bénard cell[J/OL]. PhysicalReviewLetters,2022, 128:084501. DOI: 10.1103/PhysRevLett.128.084501.
[51] REN L, TAO X, ZHANG L, et al. Flow states and heat transport in liquid metalconvection[J]. Journal of Fluid Mechanics, 2022, 951:R1.
[52] CHEN X, WANG D P, XI H D. Reduced flow reversals in turbulent convection inthe absence of corner vortices[J]. Journal of Fluid Mechanics, 2020, 891.
[53] BELKADI M, GUISLAIN L, SERGENT A, et al. Experimental and numericalshadowgraph in turbulent rayleigh–bénard convection with a rough boundary:investigation of plumes[J]. Journal of Fluid Mechanics, 2020, 895.
[54] ROSEVEARMG, GAYEN B, GRIFFITHS RW. Turbulent horizontal convectionunder spatially periodic forcing: a regime governed by interior inertia[J]. Journalof Fluid Mechanics, 2017, 831:491.
[55] XI H D, LAM S, XIA K Q. From laminar plumes to organized flows: the onsetof large-scale circulation in turbulent thermal convection[J]. Journal of FluidMechanics, 2004, 503:47-56.
[56] TURNER J. Buoyant plumes and thermals[J]. Annual Review of Fluid Mechanics,1969, 1(1):29-44.
[57] VAN KEKEN P. Evolution of starting mantle plumes: a comparison betweennumerical and laboratory models[J]. Earth and Planetary Science Letters, 1997,148(1-2):1-11.
[58] YIH C S. Laminar free convection due to a line source of heat[J]. Eos, TransactionsAmerican Geophysical Union, 1952, 33(5):669-672.
[59] BATCHELOR G. Heat convection and buoyancy effects in fluids[J]. QuarterlyJournal of the Royal Meteorological Society, 1954, 80(345):339-358.
[60] TURNER J. The ‘starting plume’in neutral surroundings[J]. Journal of FluidMechanics, 1962, 13(3):356-368.
[61] FUJII T. Theory of the steady laminar natural convectiol above a horizontal lineheat source and a point heat source[J]. International Journal of Heat and MassTransfer, 1963, 6(7):597-606.
[62] BRAND R, LAHEY F. The heated laminar vertical jet[J]. Journal of FluidMechanics, 1967, 29(2):305-315.
[63] SHLIEN D. Some laminar thermal and plume experiments[J]. The Physics ofFluids, 1976, 19(8):1089-1098.
[64] SHLIEN D. Transition of the axisymmetric starting plume cap[J]. The Physics ofFluids, 1978, 21(12):2154-2158.
[65] MORTON B, TAYLOR G I, TURNER J S. Turbulent gravitational convectionfrom maintained and instantaneous sources[J]. Proceedings of the Royal Societyof London. Series A. Mathematical and Physical Sciences, 1956, 234(1196):1-23.
[66] STOTHERS R B. Turbulent atmospheric plumes above line sources with an applicationto volcanic fissure eruptions on the terrestrial planets[J]. Journal ofAtmospheric Sciences, 1989, 46(17):2662-2670.
[67] MOSES E, ZOCCHI G, PROCACCIA I, et al. The dynamics and interaction oflaminar thermal plumes[J]. Europhysics Letters, 1991, 14(1):55.
[68] MOSES E, ZOCCHI G, LIBCHABERII A. An experimental study of laminarplumes[J]. Journal of Fluid Mechanics, 1993, 251:581-601.
[69] KAMINSKI E, JAUPART C. Laminar starting plumes in high-prandtl-numberfluids[J]. Journal of Fluid Mechanics, 2003, 478:287-298.
[70] WORSTER M G. The axisymmetric laminar plume: asymptotic solution for largeprandtl number[J]. Studies in Applied Mathematics, 1986, 75(2):139-152.
[71] DAVAILLE A, LIMARE A, TOUITOU F, et al. Anatomy of a laminar startingthermal plume at high prandtl number[J]. Experiments in Fluids, 2011, 50(2):285-300.
[72] MORTON B. Weak thermal vortex rings[J]. Journal of Fluid Mechanics, 1960, 9(1):107-118.
[73] SHLIEN D, THOMPSON D. Some experiments on the motion of an isolatedlaminar thermal[J]. Journal of Fluid Mechanics, 1975, 72(1):35-47.
[74] GRIFFITHS R. Thermals in extremely viscous fluids, including the effects oftemperature-dependent viscosity[J]. Journal of Fluid Mechanics, 1986, 166:115-138.
[75] WHITTAKER R J, LISTER J R. The self-similar rise of a buoyant thermal in veryviscous flow[J]. Journal of Fluid Mechanics, 2008, 606:295.
[76] HATTORI T, BARTOS N, NORRIS S, et al. Experimental and numerical investigationof unsteady behaviour in the near-field of pure thermal planar plumes[J].Experimental Thermal and Fluid Science, 2013, 46:139-150.
[77] GRIFFITHS R W, GAYEN B. Turbulent convection insights from small-scalethermal forcing with zero net heat flux at a horizontal boundary[J]. PhysicalReview Letters, 2015, 115(20):204301.
[78] PERA L, GEBHART B. Laminar plume interactions[J]. Journal of Fluid Mechanics,1975, 68(2):259-271.
[79] ROONEY G. Merging of two or more plumes arranged around a circle[J]. Journalof Fluid Mechanics, 2016, 796:712-731.
[80] ROONEY G. Merging of a row of plumes or jets with an application to plume risein a channel[J]. Journal of Fluid Mechanics, 2015, 771.
[81] LI S, FLYNN M. Merging of long rows of plumes: Crosswinds, multiple rows,and applications to cooling towers[J]. Physical Review Fluids, 2020, 5(9):094502.
[82] LI S, FLYNN M. Merging of two plumes from area sources with applications tocooling towers[J]. Physical Review Fluids, 2020, 5(5):054502.
[83] BERTSCH A, BONGARZONE A, YIM E, et al. Swinging jets[J]. Physical ReviewFluid, 2020, 5:110505.
[84] ATTANASI A, CAVAGNA A, DEL CASTELLO L, et al. Finite-size scaling as away to probe near-criticality in natural swarms[J]. Physical Review Letters, 2014,113(23):238102.
[85] PRIGOGINE I, LEFEVER R. Symmetry breaking instabilities in dissipative systems.ii[J]. The Journal of Chemical Physics, 1968, 48(4):1695-1700.
[86] MARCHETTI M C, JOANNY J F, RAMASWAMY S, et al. Hydrodynamics ofsoft active matter[J]. Reviews of Modern Physics, 2013, 85(3):1143.
[87] ISING E. Beitrag zur theorie des ferromagnetismus[J]. Zeitschrift Für Physik,1925, 31(1):253-258.
[88] C. R. Boids[EB/OL]. 1986. http://www.red3d.com/cwr/boids/.
[89] VICSEK T, CZIROK A, BEN-JACOB E, et al. Novel type of phase transition in asystem of self-driven particles[J]. Physical Review Letters, 1995, 75(6):1226.
[90] COUZIN I D, KRAUSE J, JAMES R, et al. Collective memory and spatial sortingin animal groups[J]. Journal of Theoretical Biology, 2002, 218(1):1-11.
[91] CUCKER F, SMALE S. Emergent behavior in flocks[J]. IEEE Transactions onAutomatic Control, 2007, 52(5):852-862.
[92] VÁSÁRHELYI G, VIRÁGH C, SOMORJAI G, et al. Outdoor flocking and formationflight with autonomous aerial robots[C]//2014 IEEE/RSJ International Conferenceon Intelligent Robots and Systems. [S.l.]: IEEE, 2014: 3866-3873.
[93] RAMASWAMY S, RAO M. Active-filament hydrodynamics: instabilities, boundaryconditions and rheology[J]. New Journal of Physics, 2007, 9(11):423.
[94] MARTIN P C, PARODI O, PERSHAN P S. Unified hydrodynamic theory forcrystals, liquid crystals, and normal fluids[J]. Physical Review A, 1972, 6(6):2401.
[95] TONER J, TU Y. Flocks, herds, and schools: A quantitative theory of flocking[J].Physical Review E, 1998, 58(4):4828.
[96] EMRICH R J. Fluid dynamics[M]. [S.l.]: Academic Press, 1981.
[97] MERZKIRCH W. Flow visualization[M]. [S.l.]: Elsevier, 2012.
[98] ANUTA P E. Spatial registration of multispectral and multitemporal digital imageryusing fast fourier transform techniques[J/OL]. IEEE Transactions on GeoscienceElectronics, 1970, 8(4):353-368. DOI: 10.1109/TGE.1970.271435.
[99] KEATING T J, WOLF P, SCARPACE F. An improved method of digital imagecorrelation[J]. Photogrammetric Engineering and Remote Sensing, 1975, 41(8):993-1002.
[100] CHU T, RANSON W, SUTTON M A. Applications of digital-image-correlationtechniques to experimental mechanics[J]. Experimental Mechanics, 1985, 25:232-244.
[101] SUTTON M A, ORTEU J J, SCHREIER H. Image correlation for shape, motionand deformation measurements: basic concepts, theory and applications[M]. [S.l.]:Springer Science & Business Media, 2009.
[102] YANGJ,BHATTACHARYAK. Combining image compression with digital imagecorrelation[J]. Experimental Mechanics, 2019, 59:629-642.
[103] DYNAMICS D. Dantec dynamics a/s[M]. [S.l.]: Nova Instruments Company,2016.
[104] BRANDT L, COLETTI F. Particle-laden turbulence: progress and perspectives[J]. Annual Review of Fluid Mechanics, 2022, 54:159-189.
[105] JANKE T, MICHAELIS D. Uncertainty quantification for ptv/lpt data and adaptivetrack filtering[C]//14th International Symposium on Particle Image Velocimetry:volume 1. [S.l.: s.n.], 2021.
[106] VOTH G A, LA PORTA A, CRAWFORD A M, et al. Measurement of particleaccelerations in fully developed turbulence[J]. Journal of Fluid Mechanics, 2002,469:121-160.
[107] GESEMANN S, HUHN F, SCHANZ D, et al. From noisy particle tracks tovelocity, acceleration and pressure fields using b-splines and penalties[C]//18thinternational symposium on applications of laser and imaging techniques to fluidmechanics, Lisbon, Portugal: volume 4. [S.l.: s.n.], 2016.
[108] GESEMANN S. Trackfit: uncertainty quantification, optimal filtering and interpolationof tracks for time-resolved lagrangian particle tracking[C]//14th InternationalSymposium on Particle Image Velocimetry: volume 1. [S.l.: s.n.], 2021.
[109] MORDANT N, CRAWFORD A M, BODENSCHATZ E. Experimental lagrangianacceleration probability density function measurement[J]. Physica D: NonlinearPhenomena, 2004, 193(1-4):245-251.
[110] KOLODNER P, TYSON J A. Microscopic fluorescent imaging of surface temperatureprofiles with 0.01 c resolution[J]. Applied Physics Letters, 1982, 40(9):782-784.
[111] TROPEA C, YARIN A L, FOSS J F, et al. Springer handbook of experimentalfluid mechanics: volume 1[M]. [S.l.]: Springer, 2007.
[112] SCHRÖDER A, SCHANZ D. 3d lagrangian particle tracking in fluid mechanics[J]. Annual Review of Fluid Mechanics, 2023, 55:511-540.
[113] GAO Z Y, LUO J H, BAO Y. Numerical study of heat-transfer in two-and quasitwo-dimensional rayleigh–bénard convection[J]. Chinese Physics B, 2018, 27(10):104702.
[114] ZHANG Y Z, SUN C, BAO Y, et al. How surface roughness reduces heat transportfor small roughness heights in turbulent rayleigh–bénard convection[J]. Journal ofFluid Mechanics, 2018, 836.
[115] BAO Y, LUO J, YE M. Parallel direct method of dns for two-dimensional turbulentrayleigh-bénard convection[J]. Journal of Mechanics, 2018, 34(2):159-166.
[116] SPIEGEL E A, VERONIS G. On the boussinesq approximation for a compressiblefluid.[J]. The Astrophysical Journal, 1960, 131:442.
[117] BUCKINGHAM E. On physically similar systems; illustrations of the use ofdimensional equations[J]. Physical Review, 1914, 4(4):345.
[118] HARLOW F H, WELCH J E. Numerical calculation of time-dependent viscousincompressible flow of fluid with free surface[J]. Physics of Fluids, 1965, 8(12):2182-2189.
[119] GONG Z, FU X. A pencil distributed direct numerical simulation solver with versatiletreatments for viscous term[J]. Computers&Mathematics with Applications,2021, 100:141-151.
[120] WESSELINGP. Principles of computational fluid dynamics: volume 29[M]. [S.l.]:Springer Science & Business Media, 2009.
[121] CHORIN A J. On the convergence of discrete approximations to the navier-stokesequations[J]. Mathematics of Computation, 1969, 23(106):341-353.calculations[J]. Journal of Computational Physics, 1970, 6(2):322-325.
[124] SCHUMANN U, SWEET R A. Fast fourier transforms for direct solution of poisson’sequation with staggered boundary conditions[J]. Journal of ComputationalPhysics, 1988, 75(1):123-137.
[125] DODDMS, FERRANTE A. A fast pressure-correction method for incompressibletwo-fluid flows[J]. Journal of Computational Physics, 2014, 273:416-434.
[126] KOLMOGOROV A N. Equations of turbulent motion in an incompressible fluid[C]//Dokl. Akad. Nauk SSSR: volume 30. [S.l.: s.n.], 1941: 299-303.
[127] BATCHELOR G K. Small-scale variation of convected quantities like temperaturein turbulent fluid part 1. general discussion and the case of small conductivity[J].Journal of Fluid Mechanics, 1959, 5(1):113-133.
[128] SHISHKINA O, STEVENS R J, GROSSMANN S, et al. Boundary layer structurein turbulent thermal convection and its consequences for the required numericalresolution[J]. New Journal of Physics, 2010, 12(7):075022.
[129] GROSSMANN S, LOHSE D. Thermal convection for large prandtl numbers[J].Physical Review Letters, 2001, 86(15):3316.
[130] GROSSMANN S, LOHSE D. Scaling in thermal convection: a unifying theory[J].Journal of Fluid Mechanics, 2000, 407:27-56.
[131] ATKINSON J, DAVIDSON P. The evolution of laminar thermals[J]. Journal ofFluid Mechanics, 2019, 878:907-931.
[132] PARSONS J, J. R., MULLIGANJ C. Onset ofNatural Convection from a SuddenlyHeated Horizontal Cylinder[J]. Journal of Heat Transfer, 1980, 102(4):636-639.
[133] BIRD R B, STEWART W E, LIGHTFOOT E N, et al. Transport phenomena[J].Journal of The Electrochemical Society, 1961, 108(3):78C.
[134] ZHOU Q, XIA K Q. Thermal boundary layer structure in turbulent rayleigh–bénard convection in a rectangular cell[J]. Journal of Fluid Mechanics, 2013, 721:199-224.
[135] PATTERSON J, IMBERGER J. Unsteady natural convection in a rectangularcavity[J]. Journal of Fluid Mechanics, 1980, 100(1):65-86.
[136] VEST C, LAWSON M. Onset of convection near a suddenly heated horizontalwire[J]. International Journal of Heat and Mass Transfer, 1972, 15(6):1281-1283.
[137] TAKESHITA T, SEGAWA T, GLAZIER J A, et al. Thermal turbulence in mercury[J]. Physical Review Letters, 1996, 76(9):1465.
[138] LI X M, HE J D, HAO P, et al. Effects of prandtl number in quasi-two-dimensionalturbulent rayleigh-bénard convection[C]//APS Division of Fluid Dynamics MeetingAbstracts. [S.l.: s.n.], 2019: H14-004.
[139] LI X, SHI Y Z,WANG K, et al. Thermally activated delayed fluorescence carbonylderivatives for organic light-emitting diodes with extremely narrow full widthat half-maximum[J]. ACS Applied Materials & Interfaces, 2019, 11(14):13472-13480.
[140] KIDA S, TAKAOKA M. Vortex reconnection[J]. Annual Review of Fluid Mechanics,1994, 26(1):169-177.
[141] HE Z, ZHANG W, JIANG H, et al. Dynamic interaction and mixing of twoturbulent forced plumes in linearly stratified ambience[J]. Journal of HydraulicEngineering, 2018, 144(12):04018072.
[142] LOU Y, HE Z, JIANG H, et al. Numerical simulation of two coalescing turbulentforced plumes in linearly stratified fluids[J]. Physics of Fluids, 2019, 31(3).
[143] CHONGKL, XIAKQ. Exploring the severely confined regime in rayleigh–bénardconvection[J]. Journal of Fluid Mechanics, 2016, 805:R4.
[144] XIAKQ,HUANGS D, XIEYC, et al. Tuning heat transport via coherent structuremanipulation: Recent advances in thermal turbulence[J]. National Science Review,2023:nwad012.
[145] MOTOKI S, KAWAHARA G, SHIMIZU M. Multi-scale steady solution forrayleigh–bénard convection[J]. Journal of Fluid Mechanics, 2021, 914:A14.
[146] HUANG S D, XIA K Q. Effects of geometric confinement in quasi-2-d turbulentrayleigh–bénard convection[J]. Journal of Fluid Mechanics, 2016, 794:639-654.
[147] CHONG K L, YANG Y, HUANG S D, et al. Confined rayleigh-bénard, rotatingrayleigh-bénard, and double diffusive convection: A unifying view on turbulenttransport enhancement through coherent structure manipulation[J]. Physical ReviewLetters, 2017, 119(6):064501.
[148] KHURANA A. Rayleigh-bénard experiment probes transition from chaos to turbulence[J]. Physics Today, 1988, 41(6):17.G, BODENSCHATZ E, HARTMANN R, et al. Aspect ratio dependenceof heat transfer in a cylindrical rayleigh-bénard cell[J]. Physical Review Letters,2022, 128(8):084501.
[151] WAGNER S, SHISHKINA O. Aspect-ratio dependency of rayleigh-bénard convectionin box-shaped containers[J]. Physics of Fluids, 2013, 25(8):085110.
[152] FILELLA A, NADAL F, SIRE C, et al. Model of collective fish behavior withhydrodynamic interactions[J]. Physical Review Letters, 2018, 120(19):198101.
[153] LIU P, ZHU H, ZENG Y, et al. Oscillating collective motion of active rotors inconfinement[J]. Proceedings of the National Academy of Sciences, 2020, 117(22):11901-11907.
[154] NAGAI K H, SUMINO Y, MONTAGNE R, et al. Collective motion of selfpropelledparticles with memory[J]. Physical Review Letters, 2015, 114(16):168001.
[155] CALLAHAM J L, LOISEAU J C, RIGAS G, et al. Nonlinear stochastic modellingwith langevin regression[J]. Proceedings of the Royal Society A, 2021, 477(2250):20210092.
[156] ZHANG Y, HUANG Y X, JIANG N, et al. Statistics of velocity and temperaturefluctuations in two-dimensional rayleigh-bénard convection[J]. Physical ReviewE, 2017, 96(2):023105.
[157] SHANG X D, TONG P, XIA K Q. Scaling of the local convective heat flux inturbulent rayleigh-bénard convection[J/OL]. Physical Review Letters, 2008, 100:244503. DOI: 10.1103/PhysRevLett.100.244503.
[158] SCHINDLER F, ECKERT S, ZÜRNER T, et al. Collapse of coherent large scaleflow in strongly turbulent liquid metal convection[J]. Physical Review Letters,2022, 128(16):164501.
[159] FUNFSCHILLING D, BROWN E, NIKOLAENKO A, et al. Heat transport byturbulent rayleigh–bénard convection in cylindrical samples with aspect ratio oneand larger[J]. Journal of Fluid Mechanics, 2005, 536:145-154.
[160] HARTMANN R, CHONG K L, STEVENS R J, et al. Heat transport enhancementin confined rayleigh-bénard convection feels the shape of the container (a)[J].Europhysics Letters, 2021, 135(2):24004.
[161] NIKOLAENKO A, BROWN E, FUNFSCHILLING D, et al. Heat transport byturbulent rayleigh–bénard convection in cylindrical cells with aspect ratio one andless[J]. Journal of Fluid Mechanics, 2005, 523:251-260.
[162] ZHANG X, ECKE R E, SHISHKINA O. Boundary zonal flows in rapidly rotatingturbulent thermal convection[J]. Journal of Fluid Mechanics, 2021, 915:A62.
[163] WEISS S, AHLERS G. Turbulent rayleigh–bénard convection in a cylindricalcontainer with aspect ratio 𝛾= 0.50 and prandtl number pr= 4.38[J]. Journal ofFluid Mechanics, 2011, 676:5-40.
[164] BAILON-CUBA J, EMRAN M S, SCHUMACHER J. Aspect ratio dependenceof heat transfer and large-scale flow in turbulent convection[J]. Journal of FluidMechanics, 2010, 655:152–173.
[165] AHLERS G, HE X, FUNFSCHILLING D, et al. Heat transport by turbulentrayleigh–bénard convection for pr 0.8 and 3× 1012 ra 1015: aspect ratio 𝛾= 0.50[J]. New Journal of Physics, 2012, 14(10):103012.