Turbulent Rayleigh-B\'enard convection under strong non-Oberbeck-Boussinesq conditions

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Turbulent Rayleigh-Bénard convection under strong non-Oberbeck-Boussinesq conditions

Hiufai Yik, Valentina Valori, and Stephan Weiss

Phys. Rev. Fluids 5, 103502 – Published 20 October 2020

Abstract

We report on Rayleigh-Bénard convection with strongly varying fluid properties experimentally and theoretically. Using pressurized sulfur-hexafluoride ( ${SF}_{6}$ ) above its critical point, we are able to make measurements at mean temperatures ( $T_{m}$ ) and pressures ( $P_{m}$ ) along Prandtl-number isolines in the ( $T, P$ ) parameter space. This allows us to keep the mean Rayleigh- ( ${Ra}_{m}$ ) and Prandtl number ( $\Pr_{m}$ ) constant while changing the temperature dependences of the fluid properties independently, e.g., probing the liquidlike or gaslike region that are left and right of the supercritical isochore. Hence, non-Oberbeck-Boussinesq (NOB) effects can be measured and analyzed cleanly. We measure the temperature at midheight ( $T_{c}$ ) as well as the global vertical heat flux. We observe a significant heat transport enhancement of up to $112 %$ under strong NOB conditions. Furthermore, we develop a theoretical model for the global vertical heat flux based on ideas of Grossmann and Lohse (GL) in OB systems, adjusted for nonconstant fluid properties. In this model, the NOB effects influence the boundary layer and hence $T_{c}$ , but the change of the heat flux is predominantly due to a change of the fluid properties in the bulk, in particular the heat capacity $c_{p}$ and density $ρ$ . Predictions from our model are consistent with our experimental results as well as with previous measurements carried out in pressurized ethane and cryogenic helium.

Received 29 May 2020
Accepted 10 September 2020

DOI:https://doi.org/10.1103/PhysRevFluids.5.103502

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Rayleigh-Bénard convection Structure & turbulence of boundary layers Turbulence

Fluid Dynamics

Authors & Affiliations

Hiufai Yik ^1,*, Valentina Valori ^2,3,4, and Stephan Weiss ^1,5,†

¹Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
²Department of Radiation Science and Technology, Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands
³Process and Energy Department, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands
⁴Institut für Thermo-und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
⁵Max Planck – University of Twente Center for Complex Fluid Dynamics

^*hyik@ds.mpg.de
^†stephan.weiss@ds.mpg.de

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Vol. 5, Iss. 10 — October 2020

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Figure 1
(a) Pressure-temperature phase space for ${SF}_{6}$ with the vapor pressure (solid white line), the supercritical isochore (white dashed line), and different isocontour lines of $Pr$ (dashed line). Their color represents ${Pr}_{m} = 3.5$ (blue), 3.0 (green), and 1.5 (orange). Horizontal lines indicate the temperature range ${T_{t}, T_{b}}$ at which experiments were conducted. For all measurements it was ${Ra}_{m} = 2.8 \times 10^{11}$ . (b) $P − T$ phase space with the lines of extreme values, i.e., where $\partial α / \partial T = 0$ , $\partial ν / \partial T = 0$ , $\partial λ / \partial T = 0$ , $\partial κ / \partial T = 0$ , $\partial c_{p} / \partial T = 0$ , as well as the supercritical isochore $ρ = ρ^{*}$ . These lines separate the liquidlike from the gaslike region. We also show the isolines of $Pr = 1.5$ , 3.0, and 3.5 as gray lines for comparison.
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Figure 2
Normalized fluid properties $\tilde{X}$ (see legend) as a function of the normalized temperature $θ$ for four representative experimental runs (i.e., $T_{m}, P_{m}$ ) and ${Pr}_{m} = 3.5$ . Vertical dash-dotted line: experimentally measured $θ_{c}$ . Horizontal dotted line: $\tilde{X} (θ) = 1$ , i.e., the OB case. Inset: zoomed-in plot showing $\tilde{ρ}, \tilde{Λ}, \tilde{μ}$ . (a) Gaslike fluid at $T_{m} = 50.79^{\circ} C$ and $P_{m} = 40.47$ bar. (b) Liquidlike fluid at $T_{m} = 50.789^{\circ} C$ , $P_{m} = 45.79$ bar. (c) Fluid with the most OB-like properties, i.e., slightly above the critical isochore at $T_{m} = 58.98^{\circ} C$ and $P_{m} = 49.41$ bar. (d) Mostly liquidlike fluid at $T_{m} = 45.26^{\circ} C$ and $P_{m} = 40.56$ bar.
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Figure 3
Experimental results. In the left column (a), (c), (e), the measured center temperature (black crosses) is plotted with a prediction from the extended-boundary-layer theory (cyan line), the virtual-cells model (red dashed-dotted line), and the mixing zone model (gray dashed line). In the right column (b), (d), (f), measured ${Nu}_{m}$ (black cross) and real Nusselt number (white dot) are plotted with prediction from the proposed extended GL model by using the boundary layer predicted $θ_{c}$ (blue solid line) and using measured $θ_{c}$ (green dashed line). The standard GL model is also plotted as the OB reference (black dotted line). The vertical line marks the measurement at which NOB effects are smallest ( $θ_{c} = 0.5$ ). The shaded region shows the sensitivity of $T_{c}$ measurement by displaying the effect of a $\pm 50$ mK error. Note that the Nu shown here are already corrected by 10% so that the most OB-like measurements coincide with the GL prediction.
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Figure 4
(a) NOB Nusselt number enhancement in gaseous ethane at $T_{m} = 40^{\circ} C$ plotted against the mean Rayleigh number. Measurement data for $P_{m} = 1.104 P_{ethane}^{*}$ (red triangles), $P_{m} = 1.062 P_{ethane}^{*}$ (teal diamonds), and $P_{m} = 0.991 P_{ethane}^{*}$ (gray squares) are adapted from [19]. Model predictions by the proposed model for $P_{m} = 1.104 P_{ethane}^{*}$ (red dotted line), $P_{m} = 1.062 P_{ethane}^{*}$ (teal dashed line), and $P_{m} = 0.991 P_{ethane}^{*}$ (gray dashed-dotted line) are plotted for comparison. (b) Compensated mean Nusselt number in cryogenic helium vs mean Rayleigh number measured by [23] (black cross) and our model prediction (red dots). Dotted line: GL prediction for OB reference. The ${Ra}_{m}$ and ${Nu}_{m}$ have been recalculated using refprop 9.1 [29].
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Physical Review Fluids

Turbulent Rayleigh-Bénard convection under strong non-Oberbeck-Boussinesq conditions

Hiufai Yik, Valentina Valori, and Stephan Weiss

Phys. Rev. Fluids 5, 103502 – Published 20 October 2020

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