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Ahlers G, Bodenschatz E, Hartmann R, He X, Lohse D, Reiter P, Stevens RJAM, Verzicco R, Wedi M, Weiss S, Zhang X, Zwirner L, Shishkina O. Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell. PHYSICAL REVIEW LETTERS 2022; 128:084501. [PMID: 35275677 DOI: 10.1103/physrevlett.128.084501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2021] [Accepted: 01/13/2022] [Indexed: 06/14/2023]
Abstract
While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra_{c,Γ}∼Ra_{c,∞}(1+CΓ^{-2})^{2}, with C≲1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→Ra_{ℓ}≡Ra[Γ^{2}/(C+Γ^{2})]^{3/2} collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Ra_{u,Γ}∼[Γ^{2}/(C+Γ^{2})]^{-3/2} in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ^{-3}) must be achieved to observe the ultimate regime.
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Affiliation(s)
- Guenter Ahlers
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Eberhard Bodenschatz
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- Max Planck-University of Twente Center for Complex Fluid Dynamics, 7500 AE Enschede, Netherlands
- Institute for the Dynamics of Complex Systems, Georg-August-University Göttingen, 37073 Göttingen, Germany
- Laboratory of Atomic and Solid-State Physics and Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA
| | - Robert Hartmann
- Physics of Fluids Group, J. M. Burgers Center for Fluid Dynamics and MESA+ Institute, University of Twente, 7500 AE Enschede, Netherlands
| | - Xiaozhou He
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen, 518055 China
| | - Detlef Lohse
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- Physics of Fluids Group, J. M. Burgers Center for Fluid Dynamics and MESA+ Institute, University of Twente, 7500 AE Enschede, Netherlands
- Max Planck-University of Twente Center for Complex Fluid Dynamics, 7500 AE Enschede, Netherlands
| | - Philipp Reiter
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Richard J A M Stevens
- Physics of Fluids Group, J. M. Burgers Center for Fluid Dynamics and MESA+ Institute, University of Twente, 7500 AE Enschede, Netherlands
| | - Roberto Verzicco
- Physics of Fluids Group, J. M. Burgers Center for Fluid Dynamics and MESA+ Institute, University of Twente, 7500 AE Enschede, Netherlands
- Dipartimento di Ingegneria Industriale, University of Rome "Tor Vergata," Via del Politecnico 1, Roma 00133, Italy
- Gran Sasso Science Institute-Viale F. Crispi, 767100 L'Aquila, Italy
| | - Marcel Wedi
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Stephan Weiss
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- Max Planck-University of Twente Center for Complex Fluid Dynamics, 7500 AE Enschede, Netherlands
| | - Xuan Zhang
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Lukas Zwirner
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Olga Shishkina
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
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Zhang Y, Huang YX, Jiang N, Liu YL, Lu ZM, Qiu X, Zhou Q. Statistics of velocity and temperature fluctuations in two-dimensional Rayleigh-Bénard convection. Phys Rev E 2017; 96:023105. [PMID: 28950509 DOI: 10.1103/physreve.96.023105] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2017] [Indexed: 11/07/2022]
Abstract
We investigate fluctuations of the velocity and temperature fields in two-dimensional (2D) Rayleigh-Bénard (RB) convection by means of direct numerical simulations (DNS) over the Rayleigh number range 10^{6}≤Ra≤10^{10} and for a fixed Prandtl number Pr=5.3 and aspect ratio Γ=1. Our results show that there exists a counter-gradient turbulent transport of energy from fluctuations to the mean flow both locally and globally, implying that the Reynolds stress is one of the driving mechanisms of the large-scale circulation in 2D turbulent RB convection besides the buoyancy of thermal plumes. We also find that the viscous boundary layer (BL) thicknesses near the horizontal conducting plates and near the vertical sidewalls, δ_{u} and δ_{v}, are almost the same for a given Ra, and they scale with the Rayleigh and Reynolds numbers as ∼Ra^{-0.26±0.03} and ∼Re^{-0.43±0.04}. Furthermore, the thermal BL thickness δ_{θ} defined based on the root-mean-square (rms) temperature profiles is found to agree with Prandtl-Blasius predictions from the scaling point of view. In addition, the probability density functions of turbulent energy ɛ_{u^{'}} and thermal ɛ_{θ^{'}} dissipation rates, calculated, respectively, within the viscous and thermal BLs, are found to be always non-log-normal and obey approximately a Bramwell-Holdsworth-Pinton distribution first introduced to characterize rare fluctuations in a confined turbulent flow and critical phenomena.
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Affiliation(s)
- Yang Zhang
- Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
| | - Yong-Xiang Huang
- State Key Laboratory of Marine Environmental Science, Xiamen University, Xiamen 361102, China
| | - Nan Jiang
- Department of Mechanics, Tianjin University, Tianjin 300072, China
| | - Yu-Lu Liu
- Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China.,School of Science, Shanghai Institute of Technology, Shanghai 200235, China
| | - Zhi-Ming Lu
- Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
| | - Xiang Qiu
- School of Science, Shanghai Institute of Technology, Shanghai 200235, China
| | - Quan Zhou
- Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
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Abstract
Boiling is an extremely effective way to promote heat transfer from a hot surface to a liquid due to numerous mechanisms, many of which are not understood in quantitative detail. An important component of the overall process is that the buoyancy of the bubble compounds with that of the liquid to give rise to a much-enhanced natural convection. In this article, we focus specifically on this enhancement and present a numerical study of the resulting two-phase Rayleigh-Bénard convection process in a cylindrical cell with a diameter equal to its height. We make no attempt to model other aspects of the boiling process such as bubble nucleation and detachment. The cell base and top are held at temperatures above and below the boiling point of the liquid, respectively. By keeping this difference constant, we study the effect of the liquid superheat in a Rayleigh number range that, in the absence of boiling, would be between 2 × 10(6) and 5 × 10(9). We find a considerable enhancement of the heat transfer and study its dependence on the number of bubbles, the degree of superheat of the hot cell bottom, and the Rayleigh number. The increased buoyancy provided by the bubbles leads to more energetic hot plumes detaching from the cell bottom, and the strength of the circulation in the cell is significantly increased. Our results are in general agreement with recent experiments on boiling Rayleigh-Bénard convection.
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