Breakdown of the large-scale circulation in $\ensuremath{\Gamma}=1/2$ rotating Rayleigh-B\'enard flow

Breakdown of the large-scale circulation in $Γ = 1 / 2$ rotating Rayleigh-Bénard flow

Richard J. A. M. Stevens, Herman J. H. Clercx, and Detlef Lohse

Phys. Rev. E 86, 056311 – Published 20 November 2012

Abstract

Experiments and simulations of rotating Rayleigh-Bénard convection in cylindrical samples have revealed an increase in heat transport with increasing rotation rate. This heat transport enhancement is intimately related to a transition in the turbulent flow structure from a regime dominated by a large-scale circulation (LSC), consisting of a single convection roll, at no or weak rotation to a regime dominated by vertically aligned vortices at strong rotation. For a sample with an aspect ratio $Γ = D / L = 1$ ( $D$ is the sample diameter and $L$ is its height) the transition between the two regimes is indicated by a strong decrease in the LSC strength. In contrast, for $Γ = 1 / 2$ , Weiss and Ahlers [J. Fluid Mech. 688, 461 (2011)] revealed the presence of a LSC-like sidewall temperature signature beyond the critical rotation rate. They suggested that this might be due to the formation of a two-vortex state, in which one vortex extends vertically from the bottom into the sample interior and brings up warm fluid while another vortex brings down cold fluid from the top; this flow field would yield a sidewall temperature signature similar to that of the LSC. Here we show by direct numerical simulations for $Γ = 1 / 2$ and parameters that allow direct comparison with experiment that the spatial organization of the vertically aligned vortical structures in the convection cell do indeed yield (for the time average) a sinusoidal variation of the temperature near the sidewall, as found in the experiment. This is also the essential and nontrivial difference with the $Γ = 1$ sample, where the vertically aligned vortices are distributed randomly.

Received 25 June 2012

DOI:https://doi.org/10.1103/PhysRevE.86.056311

Authors & Affiliations

Richard J. A. M. Stevens^1,*, Herman J. H. Clercx², and Detlef Lohse³

¹Department of Science and Technology and J.M. Burgers Center for Fluid Dynamics, University of Twente, Post Office Box 217, 7500 AE Enschede, The Netherlands
²Department of Applied Mathematics, University of Twente, Enschede, The Netherlands
³Department of Physics and J.M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, Post Office Box 513, 5600 MB Eindhoven, The Netherlands

^*Current address: Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA.

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Vol. 86, Iss. 5 — November 2012

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Figure 1
(a) Sketch of a single roll state (left) and a double roll state (right) of the large scale circulation in a $Γ = 1 / 2$ sample. (b, c) Visualization of the temperature field in the horizontal midplane of a cylindrical convection cell for $Ra = 2.91 \times 10^{8}$ and $Pr = 4.38$ in a $Γ = 1$ sample [5]. The red (light gray) and blue (dark gray) areas indicate warm and cold fluid, respectively. (b) At a slow rotation rate ( $1 / Ro = 0.278$ ), and below the transition: The warm up-flow (red [light gray]) and cold down-flow (blue [dark gray]) reveal the existence of a single convection roll superimposed upon turbulent fluctuations. (c) At a somewhat larger rotation rate ( $1 / Ro = 0.775$ ), above the transition: Here, the vertically aligned vortices cause disorder on a smaller length scale in the horizontal plane.Reuse & Permissions
Figure 2
(a) The heat transfer enhancement with respect to the nonrotating case, $Nu (1 / Ro) / Nu (0)$ , and (b) the temper-ature gradient at the sidewall as function of the rotation rate 1/Ro for $Ra = 4.52 \times 10^{9}$ and $Pr = 4.38$ in a $Γ = 1 / 2$ sample. The experi-mental data from Weiss and Ahlers [23] are indicated by the blue open squares and the simulation results with the red solid circles. The vertical dashed lines indicates the position of the transition (at 1/Ro $_{c}$ $\approx$ 0.86) [21].Reuse & Permissions
Figure 3
The values of (a) ${\bar{S}}_{b}$ (at $0.25 z / L$ ), (b) ${\bar{S}}_{m}$ (at $0.50 z / L$ ), and (c) ${\bar{S}}_{t}$ (at $0.75 z / L$ ) as function of 1/Ro. The experimental data from Weiss and Ahlers [23] are indicated by blue open squares and the simulation results with red solid circles. The experimental and simulation results are for $Ra = 4.52 \times 10^{9}$ and $Pr = 4.38$ in a $Γ = 1 / 2$ sample. For the simulation data $\bar{S}$ is based on the temperature measurements of 64 azimuthally equally spaced probes outside the sidewall boundary layer and for the experimental data it is based on temperature measurements of 8 probes embedded in the sidewall. The vertical dashed line indicates the position of 1/Ro $_{c}$ $\approx$ 0.86 [21].Reuse & Permissions
Figure 4
The time-averaged temperature amplitudes $〈 δ_{k} 〉$ for all three levels $k \in {b, m, t}$ , with the symbols representing $b$ (red circles), $m$ (black squares), and $t$ (blue triangles) as function of 1/Ro for $Γ = 1 / 2$ and $Ra = 4.52 \times 10^{9}$ . Vertical dashed line as in Fig. 3.Reuse & Permissions
Figure 5
Flow visualization for $Ra = 4.52 \times 10^{9}$ , $Pr = 4.38$ , and $1 / Ro = 3.33$ in an aspect ratio $Γ = 1 / 2$ sample. The color map for the horizontal planes at $z / L = 0.25$ , $z / L = 0.50$ , and $z / L = 0.75$ is different in panels (a) and (b) and is indicated on the left- and right-hand sides, respectively. The red temperature isosurface, originating from the bottom, indicates the region $Δ ≳ 0.65$ , whereas the blue isosurface, originating from the top, indicates the region $Δ ≲ 0.35$ . Corresponding movies can be found in the supplementary material [41].Reuse & Permissions
Figure 6
Flow visualization for $Ra = 4.52 \times 10^{9}$ , $Pr = 4.38$ , and $1 / Ro = 3.33$ in an aspect ratio $Γ = 1 / 2$ sample. The top row shows three-dimensional temperature isosurfaces for two different time instances where red regions, originating from the bottom, indicate $Δ ≳ 0.65$ and the blue isosurfaces, originating from the top, indicate the region $Δ ≲ 0.35$ ; see Fig. 5. The bottom row shows a visualization of the vortices, based on the $Q_{3 D}$ criterion. Note that the red (blue) regions indicated by the temperature isosurface indeed correspond to vortex regions.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Breakdown of the large-scale circulation in $Γ = 1 / 2$ rotating Rayleigh-Bénard flow

Richard J. A. M. Stevens, Herman J. H. Clercx, and Detlef Lohse

Phys. Rev. E 86, 056311 – Published 20 November 2012

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Physical Review E

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Breakdown of the large-scale circulation in Γ=1/2 rotating Rayleigh-Bénard flow

Richard J. A. M. Stevens, Herman J. H. Clercx, and Detlef Lohse

Phys. Rev. E 86, 056311 – Published 20 November 2012

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Breakdown of the large-scale circulation in $Γ = 1 / 2$ rotating Rayleigh-Bénard flow