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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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Curbelo J, Lopez JM, Mancho AM, Marques F. Confined rotating convection with large Prandtl number: centrifugal effects on wall modes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:013019. [PMID: 24580332 DOI: 10.1103/physreve.89.013019] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2013] [Indexed: 06/03/2023]
Abstract
Thermal convection in a rotating cylinder with a radius-to-height aspect ratio of Γ=4 for fluids with large Prandtl number is studied numerically. Centrifugal buoyancy effects are investigated in a regime where the Coriolis force is relatively large and the onset of thermal convection is in the so-called wall modes regime, where pairs of hot and cold thermal plumes ascend and descend in the cylinder sidewall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot and cold plume pairs. In our numerical study, we use the physical parameters corresponding to aqueous mixtures of glycerine with mass concentration in the range of 60%-90% glycerine and a Rayleigh number range that extends from the threshold for wall modes up to values where the bulk fluid region is also convecting. The study shows that for the range of Rayleigh numbers considered, the local variations in viscosity due to temperature variation in the flow are negligible. However, the mean viscosity, which varies faster than exponentially with variations in the percentage of glycerine, leads to a faster than exponential increase in the Froude number for a fixed Coriolis force, and hence an enhancement of the centrifugal buoyancy effects with significant dynamical consequences, which are detailed.
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Affiliation(s)
- Jezabel Curbelo
- Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera, 15, Madrid 28049, Spain and Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid 28049, Spain
| | - Juan M Lopez
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287, USA
| | - Ana M Mancho
- Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera, 15, Madrid 28049, Spain
| | - Francisco Marques
- Department of Física Aplicada, Universitat Politècnica de Catalanya, Girona s/n, Mòdul B4 Campus Nord, 08034 Barcelona, Spain
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