Rayleigh­Bénard 热对流中惯性小球的拉格朗日动力学及其影响

颗粒负载流广泛存在于自然界以及工程应用中，如沙尘暴、海洋浮标、催化剂 粒子的混合与输运等。其中，大尺寸惯性颗粒在湍流中的动力学及其对流动的影响，目前还有待研究。本文基于经典 Rayleigh­Bénard 对流系统，制备了一种能跟随流体运动的厘米尺度的惯性小球，通过 PIV 测量技术和光影显示技术，实验研究了大尺寸惯性小球在经典 Rayleigh­Béard 热对流系统中的拉格朗日动力学行为， 以及不同体积分数 𝜙 下惯性小球对流动和系统传热的影响。本文的主要结论如下。 在 𝑅𝑎 数 8.8 × 10⁸ ∼ 8.8 × 10⁹ 范围内，单个惯性小球的均方位移 (MSD) 在较长的时间里满足弹道扩散，并在到达由于空间受限所引起的平台区后表现出一定程度的振荡，相应的特征时间与系统中大尺度流动结构的周期一致。这种周期在其它物理量如自关联函数中亦有体现。拉格朗日速度、加速度的概率密度分布分别为高斯分布和拉伸的指数分布，与通过示踪粒子得到的结果一致。此外，我们还发现欧拉视角和拉格朗日视角刻画的速度场在空间分布和统计特性上基本一致， 但在强度大小上有一定差别。 改变惯性小球的体积分数，发现 𝜙 在 0 ∼ 10.4％范围内时，惯性小球的拉格朗日动力学基本不变。当 𝜙=17.2％、28％时，贯穿全场的 Rayleigh­-Bénard 系统大尺度结构消失，出现局部的流动结构，相应的 MSD 弹道扩散特征消失，速度的概 率密度函数也开始偏离高斯分布。让人意外的是，当惯性小球在跟随流体运动时， 系统传热效率几乎不受小球体积分数的影响；而当小球沉积在系统底部时，传热效率则与其堆积面积有关。 这些结果为颗粒负载流的相关研究，尤其是大尺寸惯性颗粒在湍流中的动力学及潜在应用 (如海洋浮标观测技术)，提供了一定的实验依据和启示。

[1] MATHAI V, LOHSE D, SUN C. Bubbly and buoyant particle–laden turbulent flows[J]. Annual Review of Condensed Matter Physics, 2020, 11: 529­559.

[2] SHAW R A. Particle­turbulence interactions in atmospheric clouds[J]. Annual Review of Fluid Mechanics, 2003, 35(1): 183­227.

[3] MARTIN D, NOKES R. A fluid­dynamical study of crystal settling in convecting magmas[J]. Journal of Petrology, 1989, 30(6): 1471­1500.

[4] MEAD K S, DENNY M W. The effects of hydrodynamic shear stress on fertilization and earlydevelopment of the purple sea urchin Strongylocentrotus purpuratus[J]. The Biological Bulletin,1995, 188(1): 46­56.

[5] BOEHM M T, AYLOR D E, SHIELDS E J. Maize pollen dispersal under convective conditions [J]. Journal of Applied Meteorology and Climatology, 2008, 47(1): 291­307.

[6] BROWN J K, HOVMØLLER M S. Aerial dispersal of pathogens on the global and continental scales and its impact on plant disease[J]. Science, 2002, 297(5581): 537­541.

[7] SCHWAIGER H F, DENLINGER R P, MASTIN L G. Ash3d: A finite­volume, conservative numerical model for ash transport and tephra deposition[J]. Journal of Geophysical Research: Solid Earth, 2012, 117(B4).

[8] OUELLETTE N T, O’MALLEY P, GOLLUB J P. Transport of finite­sized particles in chaotic flow[J]. Physical Review Letters, 2008, 101(17): 174504.

[9] TOSCHI F, BODENSCHATZ E. Lagrangian properties of particles in turbulence[J]. Annual Review of Fluid Mechanics, 2009, 41: 375­404.

[10] GUHA A. Transport and deposition of particles in turbulent and laminar flow[J]. Annual Review of Fluid Mechanics, 2008, 40: 311­341.

[11] DAVIS R E. Lagrangian ocean studies[J]. Annual Review of Fluid Mechanics, 1991, 23(1):43­64.

[12] GOULD W J. From swallow floats to Argo—The development of neutrally buoyant floats[J].Deep Sea Research Part II: Topical Studies in Oceanography, 2005, 52(3­4): 529­543.

[13] WONG A P, WIJFFELS S E, RISER S C, et al. Argo data 1999–2019: Two million temperature salinity profiles and subsurface velocity observations from a global array of profiling floats[J].Frontiers in Marine Science, 2020, 7: 700.

[14] AHLERS G, GROSSMANN S, LOHSE D. Heat transfer and large scale dynamics in turbulentRayleigh­Bénard convection[J]. Reviews of Modern Physics, 2009, 81(2): 503.

[15] XIA K Q. Current trends and future directions in turbulent thermal convection[J]. Theoreticaland Applied Mechanics Letters, 2013, 3(5): 052001.

[16] CHILLÀ F, SCHUMACHER J. New perspectives in turbulent Rayleigh­Bénard convection[J].The European Physical Journal E, 2012, 35: 1­25.

[17] BODENSCHATZ E, PESCH W, AHLERS G. Recent developments in Rayleigh­Bénard con vection[J]. Annual Review of Fluid Mechanics, 2000, 32(1): 709­778.

[18] LOHSE D, XIA K Q. Small­scale properties of turbulent Rayleigh­Bénard convection[J]. An nual Review of Fluid Mechanics, 2010, 42: 335­364.

[19] BÉNARD H. Les tourbillons cellulaires dans une nappe liquide[J]. Revue Gen. Sci. Pure Appl.,1900, 11: 1261­1271.

[20] RAYLEIGH L. LIX. On convection currents in a horizontal layer of fluid, when the higher tem perature is on the under side[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1916, 32(192): 529­546.

[21] OBERBECK A. Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen[J]. Annalen der Physik, 1879, 243(6): 271­292.

[22] BOUSSINESQ J. Théorie analytique de la chaleur: mise en harmonie avec la thermodynamiqueet avec la théorie mécanique de la lumière: volume 2[M]. Gauthier­Villars, 1903.

[23] CASTAING B, GUNARATNE G, HESLOT F, et al. Scaling of hard thermal turbulence in Rayleigh­Bénard convection[J]. Journal of Fluid Mechanics, 1989, 204: 1­30.

[24] ZHOU Q, SUN C, XIA K Q. Morphological evolution of thermal plumes in turbulent Rayleigh Bénard convection[J]. Physical Review Letters, 2007, 98(7): 074501.

[25] SHANG X D, QIU X L, TONG P, et al. Measured local heat transport in turbulent Rayleigh Bénard convection[J]. Physical Review Letters, 2003, 90(7): 074501.

[26] XI H D, LAM S, XIA K Q. From laminar plumes to organized flows: the onset of large­scale circulation in turbulent thermal convection[J]. Journal of Fluid Mechanics, 2004, 503: 47­56.

[27] LI X M, HE J D, TIAN Y, et al. Effects of Prandtl number in quasi­two­dimensional Rayleigh–Bénard convection[J]. Journal of Fluid Mechanics, 2021, 915: A60.

[28] HE J C, FANG M W, GAO Z Y, et al. Effects of Prandtl number in two­dimensional turbulent convection[J]. Chinese Physics B, 2021, 30(9): 094701.

[29] ZHANG J, CHILDRESS S, LIBCHABER A. Non­Boussinesq effect: Thermal convection with broken symmetry[J]. Physics of Fluids, 1997, 9(4): 1034­1042.

[30] CIONI S, CILIBERTO S, SOMMERIA J. Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number[J]. Journal of Fluid Mechanics,1997, 335: 111­140.

[31] LAVEZZO V, CLERCX H, TOSCHI F. Role of thermal plumes on particle dispersion in a turbulent Rayleigh­Bénard cell[C]//Journal of Physics: Conference Series: volume 318. IOPPublishing, 2011: 052011.

[32] BROWN E, AHLERS G. Large­scale circulation model for turbulent Rayleigh­Bénard convection[J]. Physical Review Letters, 2007, 98(13): 134501.

[33] FUNFSCHILLING D, BROWN E, AHLERS G. Torsional oscillations of the large­scale circulation in turbulent Rayleigh–Bénard convection[J]. Journal of Fluid Mechanics, 2008, 607:119­139.

[34] XI H D, ZHOU S Q, ZHOU Q, et al. Origin of the temperature oscillation in turbulent thermal convection[J]. Physical Review Letters, 2009, 102(4): 044503.

[35] XI H D, XIA K Q. Cessations and reversals of the large­scale circulation in turbulent thermal convection[J]. Physical Review E, 2007, 75(6): 066307.

[36] XI H D, ZHOU Q, XIA K Q. Azimuthal motion of the mean wind in turbulent thermal convec tion[J]. Physical Review E, 2006, 73(5): 056312.

[37] WANG Y, LAI P Y, SONG H, et al. Mechanism of large­scale flow reversals in turbulent thermal convection[J]. Science Advances, 2018, 4(11): eaat7480.

[38] SUGIYAMA K, NI R, STEVENS R J, et al. Flow reversals in thermally driven turbulence[J].Physical Review Letters, 2010, 105(3): 034503.

[39] CHEN X, HUANG S D, XIA K Q, et al. Emergence of substructures inside the large­scale circulation induces transition in flow reversals in turbulent thermal convection[J]. Journal of Fluid Mechanics, 2019, 877: R1.

[40] XIA S N, WAN Z H, LIU S, et al. Flow reversals in Rayleigh–Bénard convection with non-Oberbeck–Boussinesq effects[J]. Journal of Fluid Mechanics, 2016, 798: 628­642.

[41] WANG Q, XIA S N, WANG B F, et al. Flow reversals in two­dimensional thermal convection in tilted cells[J]. Journal of Fluid Mechanics, 2018, 849: 355­372.

[42] XIAO Y, TAO J, MA X, et al. Oscillating convection and reversal flow in connected cavities [J]. Physical Review E, 2018, 98(6): 063109.

[43] XIA Z, SHI Y, CAI Q, et al. Multiple states in turbulent plane Couette flow with spanwise rotation[J]. Journal of Fluid Mechanics, 2018, 837: 477­490.

[44] XIE Y C, DING G Y, XIA K Q. Flow topology transition via global bifurcation in thermally driven turbulence[J]. Physical Review Letters, 2018, 120(21): 214501.

[45] ZWIRNER L, TILGNER A, SHISHKINA O. Elliptical instability and multiple­roll flow modes of the large­scale circulation in confined turbulent Rayleigh­Bénard convection[J]. Physical Review Letters, 2020, 125(5): 054502.

[46] LIU S, JIANG L, CHONG K L, et al. From Rayleigh–Bénard convection to porous­media convection: how porosity affects heat transfer and flow structure[J]. Journal of Fluid Mechanics,2020, 895: A18.

[47] WANG Q, VERZICCO R, LOHSE D, et al. Multiple states in turbulent large­aspect­ratio ther mal convection: What determines the number of convection rolls?[J]. Physical Review Letters,2020, 125(7): 074501.

[48] MALKUS W V. The heat transport and spectrum of thermal turbulence[J]. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1954, 225(1161):196­212.

[49] KRAICHNAN R H. Turbulent thermal convection at arbitrary Prandtl number[J]. The Physics of Fluids, 1962, 5(11): 1374­1389.

[50] SHRAIMAN B I, SIGGIA E D. Heat transport in high­Rayleigh­number convection[J]. Phys ical Review A, 1990, 42(6): 3650.

[51] GROSSMANN S, LOHSE D. Scaling in thermal convection: a unifying theory[J]. Journal of Fluid Mechanics, 2000, 407: 27­56.

[52] GROSSMANN S, LOHSE D. Thermal convection for large Prandtl numbers[J]. Physical Review Letters, 2001, 86(15): 3316.

[53] GROSSMANN S, LOHSE D. Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection[J]. Physical Review E, 2002, 66(1): 016305.

[54] GROSSMANN S, LOHSE D. Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes[J]. Physics of Fluids, 2004, 16(12): 4462­4472.

[55] LANDAU L, LIFSHITZ E. Fluid Mechanics, V. 6 of Course of Theoretical Physics, 2nd English edition. Revised[M]. Pergamon Press, Oxford­New York­Beijing­Frankfurt­San Paulo­SydneyTokyo­Toronto, 1987.

[56] ZHOU Q, XIA K Q. Measured instantaneous viscous boundary layer in turbulent Rayleigh Bénard convection[J]. Physical Review Letters, 2010, 104(10): 104301.

[57] 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: 199­224.

[58] HUANG S D, KACZOROWSKI M, NI R, et al. Confinement­induced heat­transport enhance ment in turbulent thermal convection[J]. Physical Review Letters, 2013, 111(10): 104501.

[59] SCHINDLER F, ECKERT S, ZÜRNER T, et al. Collapse of coherent large scale flow in strongly turbulent liquid metal convection[J]. Physical Review Letters, 2022, 128(16): 164501.

[60] SUN C, XI H D, XIA K Q. Azimuthal symmetry, flow dynamics, and heat transport in turbulent thermal convection in a cylinder with an aspect ratio of 0.5[J]. Physical Review Letters, 2005, 95(7): 074502.

[61] SHEVKAR P P, GUNASEGARANE G, MOHANAN S K, et al. Effect of shear on coherent structures in turbulent convection[J]. Physical Review Fluids, 2019, 4(4): 043502.

[62] ZHANG L, DONG J, XIA K Q. Exploring the plume and shear effects in turbulent Rayleigh–Bénard convection with effective horizontal buoyancy under streamwise and spanwise geomet rical confinements[J]. Journal of Fluid Mechanics, 2022, 940: A37.

[63] SCAGLIARINI A, GYLFASON Á, TOSCHI F. Heat­flux scaling in turbulent Rayleigh­Bénard convection with an imposed longitudinal wind[J]. Physical Review E, 2014, 89(4): 043012.

[64] SHEN Y, TONG P, XIA K Q. Turbulent convection over rough surfaces[J]. Physical Review Letters, 1996, 76(6): 908.

[65] GOLUSKIN D, DOERING C R. Bounds for convection between rough boundaries[J]. Journal of Fluid Mechanics, 2016, 804: 370­386.

[66] ZHANG Y Z, SUN C, BAO Y, et al. How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh–Bénard convection[J]. Journal of Fluid Mechanics,2018, 836: R2.

[67] VASILIEV A, SUKHANOVSKII A. Turbulent convection in a cube with mixed thermal bound ary conditions: low Rayleigh number regime[J]. International Journal of Heat and Mass Transfer, 2021, 174: 121290.

[68] WANG B F, ZHOU Q, SUN C. Vibration­induced boundary­layer destabilization achieves massive heat­transport enhancement[J]. Science Advances, 2020, 6(21): eaaz8239.

[69] URBAN P, HANZELKA P, KRÁLIK T, et al. Thermal waves and heat transfer efficiency enhancement in harmonically modulated turbulent thermal convection[J]. Physical Review Letters, 2022, 128(13): 134502.

[70] BALACHANDAR S, EATON J K. Turbulent dispersed multiphase flow[J]. Annual Review of Fluid Mechanics, 2010, 42: 111­133.

[71] XU H, PUMIR A, BODENSCHATZ E. Lagrangian view of time irreversibility of fluid turbulence[J]. Science China Physics, Mechanics & Astronomy, 2016, 59: 1­9.

[72] WANG C, YI L, JIANG L, et al. How do the finite­size particles modify the drag in Taylor–Couette turbulent flow[J]. Journal of Fluid Mechanics, 2022, 937: A15.

[73] CALZAVARINI E, VOLK R, BOURGOIN M, et al. Acceleration statistics of finite­sized par ticles in turbulent flow: the role of Faxén forces[J]. Journal of Fluid Mechanics, 2009, 630:179­189.

[74] BRANDT L, COLETTI F. Particle­laden turbulence: progress and perspectives[J]. Annual Review of Fluid Mechanics, 2022, 54: 159­189.

[75] BABIANO A, CARTWRIGHT J H, PIRO O, et al. Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems[J]. Physical Review Letters, 2000, 84(25): 5764.

[76] QURESHI N M, BOURGOIN M, BAUDET C, et al. Turbulent transport of material particles:an experimental study of finite size effects[J]. Physical Review Letters, 2007, 99(18): 184502.

[77] VOTH G A, LA PORTA A, CRAWFORD A M, et al. Measurement of particle accelerations in fully developed turbulence[J]. Journal of Fluid Mechanics, 2002, 469: 121­160.

[78] VOLK R, CALZAVARINI E, VERHILLE G, et al. Acceleration of heavy and light particles in turbulence: comparison between experiments and direct numerical simulations[J]. Physica D:Nonlinear Phenomena, 2008, 237(14­17): 2084­2089.

[79] GASTEUIL Y, SHEW W L, GIBERT M, et al. Lagrangian temperature, velocity, and local heat flux measurement in Rayleigh­Bénard convection[J]. Physical Review Letters, 2007, 99(23):234302.

[80] SCHUMACHER J. Lagrangian dispersion and heat transport in convective turbulence[J]. Physical Review Letters, 2008, 100(13): 134502.

[81] SCHUMACHER J. Lagrangian studies in convective turbulence[J]. Physical Review E, 2009,79(5): 056301.

[82] XU A, TAO S, SHI L, et al. Transport and deposition of dilute microparticles in turbulent thermal convection[J]. Physics of Fluids, 2020, 32(8): 083301.

[83] LIU S, JIANG L, WANG C, et al. Lagrangian dynamics and heat transfer in porous­media convection[J]. Journal of Fluid Mechanics, 2021, 917: A32.

[84] TAGHIZADEH E, VALDÉS­PARADA F J, WOOD B D. Preasymptotic Taylor dispersion: evolution from the initial condition[J]. Journal of Fluid Mechanics, 2020, 889: A5.

[85] YANG W, ZHANG Y Z, WANG B F, et al. Dynamic coupling between carrier and dispersed phases in Rayleigh–Bénard convection laden with inertial isothermal particles[J]. Journal of Fluid Mechanics, 2022, 930: A24.

[86] TAKEUCHI S, MIYAMORI Y, GU J, et al. Flow reversals in particle­dispersed natural convection in a two­dimensional enclosed square domain[J]. Physical Review Fluids, 2019, 4(8):084304.

[87] JIANG L, CALZAVARINI E, SUN C. Rotation of anisotropic particles in Rayleigh–Bénard turbulence[J]. Journal of Fluid Mechanics, 2020, 901: A8.

[88] SALAZAR J P, DE JONG J, CAO L, et al. Experimental and numerical investigation of inertial particle clustering in isotropic turbulence[J]. Journal of Fluid Mechanics, 2008, 600: 245­256.

[89] YEO K, DONG S, CLIMENT E, et al. Modulation of homogeneous turbulence seeded with finite size bubbles or particles[J]. International Journal of Multiphase Flow, 2010, 36(3): 221­233.

[90] HU S Y, WANG K Z, JIA L B, et al. Enhanced heat transport in thermal convection with suspensions of rod­like expandable particles[J]. Journal of Fluid Mechanics, 2021, 928: R1.

[91] PATOČKA V, CALZAVARINI E, TOSI N. Settling of inertial particles in turbulent Rayleigh-Bénard convection[J]. Physical Review Fluids, 2020, 5(11): 114304.

[92] ALISEDA A, LASHERAS J. Preferential concentration and rise velocity reduction of bubbles immersed in a homogeneous and isotropic turbulent flow[J]. Physics of Fluids, 2011, 23(9):093301.

[93] WILL J B, KRUG D. Dynamics of freely rising spheres: the effect of moment of inertia[J].Journal of Fluid Mechanics, 2021, 927: A7.

[94] MORDANT N, CRAWFORD A M, BODENSCHATZ E. Experimental Lagrangian accelera tion probability density function measurement[J]. Physica D: Nonlinear Phenomena, 2004, 193(1­4): 245­251.

[95] LIOT O, SEYCHELLES F, ZONTA F, et al. Simultaneous temperature and velocity Lagrangian measurements in turbulent thermal convection[J]. Journal of Fluid Mechanics, 2016, 794: 655­675.

[96] LI X M, HUANG S D, NI R, et al. Lagrangian velocity and acceleration measurements in plume­rich regions of turbulent Rayleigh­Bénard convection[J]. Physical Review Fluids, 2021,6(5): 053503.

[97] BOURGOIN M, QURESHI N M, BAUDET C, et al. Turbulent transport of finite sized ma terial particles[C]//Journal of Physics: Conference Series: volume 318. IOP Publishing, 2011:012005.

[98] QURESHI N M, ARRIETA U, BAUDET C, et al. Acceleration statistics of inertial particles in turbulent flow[J]. The European Physical Journal B, 2008, 66: 531­536.

[99] XIA K Q, SUN C, ZHOU S Q. Particle image velocimetry measurement of the velocity field in turbulent thermal convection[J]. Physical Review E, 2003, 68(6): 066303.

[100] LUCCI F, FERRANTE A, ELGHOBASHI S. Is Stokes number an appropriate indicator for turbulence modulation by particles of Taylor­length­scale size?[J]. Physics of Fluids, 2011, 23(2): 025101.

[101] SHANG X D, TONG P, XIA K Q. Scaling of the local convective heat flux in turbulent Rayleigh­Bénard convection[J]. Physical Review Letters, 2008, 100(24): 244503.

[102] SHEN Y, XIA K Q, TONG P. Measured local­velocity fluctuations in turbulent convection[J]. Physical Review Letters, 1995, 75(3): 437.

[103] KACZOROWSKI M, XIA K Q. Turbulent flow in the bulk of Rayleigh–Bénard convection: small­scale properties in a cubic cell[J]. Journal of Fluid Mechanics, 2013, 722: 596­617.

[104] FUNFSCHILLING D, BROWN E, NIKOLAENKO A, et al. Heat transport by turbulentRayleigh–Bénard convection in cylindrical samples with aspect ratio one and larger[J/OL].Journal of Fluid Mechanics, 2005, 536: 145–154. DOI: 10.1017/S0022112005005057.

[105] KACZOROWSKI M, CHONG K L, XIA K Q. Turbulent flow in the bulk of Rayleigh–Bé nard convection: aspect­ratio dependence of the small­scale properties[J/OL]. Journal of Fluid Mechanics, 2014, 747: 73–102. DOI: 10.1017/jfm.2014.154.

[106] HUANG S D, XIA K Q. Effects of geometric confinement in quasi­2­D turbulent Rayleigh–Bénard convection[J/OL]. Journal of Fluid Mechanics, 2016, 794: 639–654. DOI: 10.1017/jfm.2016.181.

[107] JIANG H, ZHU X, MATHAI V, et al. Controlling heat transport and flow structures in thermal turbulence using ratchet surfaces[J]. Physical Review Letters, 2018, 120(4): 044501.