Abstract
In this paper, we investigate the Lagrangian coherent structures (LCSs) and their heat-transport mechanism in turbulent Rayleigh-Bénard (RB) convection. Direct numerical simulations (DNS) are performed in a closed square cell with Rayleigh numbers (Ra) ranging from 106 to 109 and Prandtl (Pr) number fixed at Pr = 0.7. First, our results show the power-law relationship between Nusselt number (Nu) and Ra, Nu = 0.99Ra0.30±0.02, confirming the results from previous studies. To gain insights into the material transport, LCSs are extracted using the finite-time Lyapunov exponent (FTLE) method. Interestingly, lobe structures are widely present, and we elucidate their role in transporting heat from the corner rolls to large-scale circulation. Next, the relationships between LCSs and thermal plumes are examined, and we identify two behaviors of thermal plumes: first, most plumes transport along the LCSs; second, few plumes are exposed to the bulk and subsequently mix with the turbulent background. Furthermore, we quantify the heat flux along the LCSs, which contributes to about 85% of the total flux regardless of Ra. This suggests that LCSs play a significant role in heat transport. Finally, the viscous (thermal) dissipation rate along the LCSs is quantified, which is larger than 80% (60%) of the total value, suggesting that LCSs are responsible for the large viscous and thermal dissipations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Stevens B. Atmospheric moist convection. Annu Rev Earth Planet Sci, 2005, 33: 605–643
Thorpe S A. Recent developments in the study of ocean turbulence. Annu Rev Earth Planet Sci, 2004, 32: 91–109
Incropera F P. Convection heat transfer in electronic equipment cooling. J Heat Transfer, 1988, 110: 1097–1111
Chillà F, Schumacher J. New perspectives in turbulent Rayleigh-Bénard convection. Eur Phys J E, 2012, 35: 1–25
Ahlers G, Grossmann S, Lohse D. Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev Mod Phys, 2009, 81: 503–537
Lohse D, Xia K Q. Small-scale properties of turbulent Rayleigh-Bénard convection. Annu Rev Fluid Mech, 2010, 42: 335–364
Ahlers G, Funfschilling D, Bodenschatz E. Transitions in heat transport by turbulent convection at Rayleigh numbers up to 1015. New J Phys, 2009, 11: 123001
Lam S, Shang X D, Zhou S Q, et al. Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh-Bénard convection. Phys Rev E, 2002, 65: 066306
Chillá F, Ciliberto S, Innocenti C, et al. Boundary layer and scaling properties in turbulent thermal convection, Il Nuovo Cimento D, 1993, 15: 1229–1249
Shishkina O, Wagner C. Local heat fluxes in turbulent Rayleigh-Bénard convection. Phys Fluids, 2007, 19: 085107
Stevens R J A M, Lohse D, Verzicco R. Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J Fluid Mech, 2011, 688: 31–43
Schmalzl J, Breuer M, Hansen U. On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys Lett, 2004, 67: 390–396
Malkus W V R. The heat transport and spectrum of thermal turbulence. Proc R Soc Lond A, 1954, 225: 196–212
Grossmann S, Lohse D. Scaling in thermal convection: A unifying theory. J Fluid Mech, 2000, 407: 27–56
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 Fluid Mech, 2018, 836: R2
Zhu X, Jiang L F, Zhou Q, et al. Turbulent Rayleigh-Bénard convection in an annular cell. J Fluid Mech, 2019, 869: R5
Yang J L, Zhang Y Z, Jin T C, et al. The Pr-dependence of the critical roughness height in two-dimensional turbulent Rayleigh-Bénard convection. J Fluid Mech, 2021, 911: A52
Dong D L, Wang B F, Dong Y H, et al. Influence of spatial arrangements of roughness elements on turbulent Rayleigh-Bénard convection. Phys Fluids, 2020, 32: 045114
Kadanoff L P. Turbulent heat flow: Structures and scaling. Phys Today, 2001, 54: 34–39
Krishnamurti R Howard L N. Large-scale flow generation in turbulent convection. Proc Natl Acad Sci USA, 1981, 78: 1981–1985
Qiu X L, Tong P. Large-scale velocity structures in turbulent thermal convection. Phys Rev E, 2001, 64: 036304
Xia K Q, Sun C, Zhou S Q. Particle image velocimetry measurement of the velocity field in turbulent thermal convection. Phys Rev E, 2003, 68: 066303
Xi H D, Lam S, Xia K Q. From laminar plumes to organized flows: The onset of large-scale circulation in turbulent thermal convection. J Fluid Mech, 1999, 503: 47–56
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. Phys Rev Lett, 2005, 95: 074502
Xi H D, Zhou Q, Xia K Q. Azimuthal motion of the mean wind in turbulent thermal convection. Phys Rev E, 2006, 73: 056312
Brown E, Ahlers G. Rotations and cessations of the large-scale circulation in turbulent Rayleigh-Bénard convection. J Fluid Mech, 2006, 568: 351–386
Hunt J, Wray A, Moin P. Eddies, Streams, and Convergence Zones in Turbulent Flows. Center for Turbulence Research Report CTR-S88, 1988. 193
Bailon-Cuba J, Schumacher J. Low-dimensional model of turbulent Rayleigh-Bénard convection in a Cartesian cell with square domain. Phys Fluids, 2011, 23: 077101
Podvin B, Sergent A. Proper orthogonal decomposition investigation of turbulent Rayleigh-Bénard convection in a rectangular cavity. Phys Fluids, 2012, 24: 105106
Paul S, Verma M K. Proper orthogonal decomposition vs. fourier analysis for extraction of large-scale structures of thermal convection. In: Proceedings of Advances in Computation, Modeling and Control of Transitional and Turbulent Flows. Singapore: World Scientific, 2016. 433–441
Schmid P J. Dynamic mode decomposition of numerical and experimental data. J Fluid Mech, 2010, 656: 5–28
Schmid P J, Li L, Juniper M P, et al. Applications of the dynamic mode decomposition. Theor Comput Fluid Dyn, 2011, 25: 249–259
Huang Y X, Zhou Q. Counter-gradient heat transport in two-dimensional turbulent Rayleigh-Bénard convection. J Fluid Mech, 2013, 737: R3
Green M A, Rowley C W, Haller G. Detection of Lagrangian coherent structures in three-dimensional turbulence. J Fluid Mech, 2007, 572: 111–120
Ma X Y, Tang Z Q, Jiang N. Eulerian and Lagrangian analysis of coherent structures in separated shear flow by time-resolved particle image velocimetry. Phys Fluids, 2020, 32: 065101
Gasteuil Y, Shew W L, Gibert M, et al. Lagrangian temperature, velocity, and local heat flux measurement in Rayleigh-Bénard convection. Phys Rev Lett, 2007, 99: 234302
Hadjighasem A, Farazmand M, Blazevski D, et al. A critical comparison of Lagrangian methods for coherent structure detection. Chaos, 2017, 27: 053104
Suara K, Khanarmuei M, Ghosh A, et al. Material and debris transport patterns in Moreton Bay, Australia: The influence of Lagrangian coherent structures. Sci Total Environ, 2020, 721: 137715
Karrasch D, Keller J. A geometric heat-flow theory of Lagrangian coherent structures. J Nonlinear Sci, 2020, 30: 1849–1888
Shadden S C, Dabiri J O, Marsden J E. Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids, 2006, 18: 047105
Procaccia I. Go with the flow. Nature, 2001, 409: 993–995
Haller G, Yuan G Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 2000, 147: 352–370
Belkin M, Niyogi P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 2003, 15: 1373–1396
Farazmand M, Haller G Polar rotation angle identifies elliptic islands in unsteady dynamical systems. Physica D, 2016, 315: 1–12
Mathur M, Haller G, Peacock T, et al. Uncovering the Lagrangian skeleton of turbulence. Phys Rev Lett, 2007, 98: 144502
Haller G, Sapsis T. Lagrangian coherent structures and the smallest finite-time Lyapunov exponent. Chaos, 2011, 21: 023115
Peacock T, Haller G. Lagrangian coherent structures: The hidden skeleton of fluid flows. Phys Today, 2013, 66: 41–47
Haller G Lagrangian coherent structures. Annu Rev Fluid Mech, 2015, 47: 137–162
Haller G Lagrangian coherent structures from approximate velocity data. Phys Fluids, 2002, 14: 1851–1861
Shadden S C, Lekien F, Marsden J E. Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D, 2005, 212: 271–304
He G S, Pan C, Feng L H, et al. Evolution of Lagrangian coherent structures in a cylinder-wake disturbed flat plate boundary layer. J Fluid Mech, 2016, 792: 274–306
Li S, Jiang N, Yang S Q, et al. Coherent structures over riblets in turbulent boundary layer studied by combining time-resolved particle image velocimetry (TRPIV), proper orthogonal decomposition (POD), and finite-time Lyapunov exponent (FTLE). Chin Phys B, 2018, 27: 104701
Schneide C, Stahn M, Pandey A, et al. Lagrangian coherent sets in turbulent Rayleigh-Bénard convection. Phys Rev E, 2019, 100: 053103
du Puits R, Li L, Resagk C, et al. Turbulent boundary layer in high Rayleigh number convection in air. Phys Rev Lett, 2014, 112: 124301
Zhang Y, Zhou Q, Sun C. Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh-Bénard convection. J Fluid Mech, 2017, 814: 165–184
Bao Y, Chen J, Liu B F, et al. Enhanced heat transport in partitioned thermal convection. J Fluid Mech, 2015, 784: R5
Chen J, Bao Y, Yin Z X, et al. Theoretical and numerical study of enhanced heat transfer in partitioned thermal convection. Int J Heat Mass Transfer, 2017, 115: 556–569
Zhang Y Z, Xia S N, Dong Y H, et al. An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows. J Hydrodyn, 2019, 31: 1159–1169
Sugiyama K, Ni R Stevens R J A M, et al. Flow reversals in thermally driven turbulence. Phys Rev Lett, 2010, 105: 034503
Zhang Y, Huang Y X, Jiang N, et al. Statistics of velocity and temperature fluctuations in two-dimensional Rayleigh-Bénard convection. Phys Rev E, 2017, 96: 023105
Wang B F, Zhou Q, Sun C. Vibration-induced boundary-layer destabilization achieves massive heat-transport enhancement. Sci Adv, 2020, 6: eaaz8239
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11732010, 92052201, 12072185, 91952102, 12032016).
Rights and permissions
About this article
Cite this article
Cheng, H., Shen, J., Zhang, Y. et al. Lagrangian coherent structures and their heat-transport mechanism in the turbulent Rayleigh-Bénard convection. Sci. China Technol. Sci. 65, 966–976 (2022). https://doi.org/10.1007/s11431-021-1970-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11431-021-1970-8