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Experiments on rapidly rotating convection: the role of the Prandtl number

Hannah M. Clercx, Rudie P. J. Kunnen

TL;DR

This work investigates how heat transfer in rapidly rotating convection depends on the Prandtl number $Pr$ within the transition between rotation-affected and geostrophic convection. Using the TROCONVEX apparatus, the authors vary $Pr$ from $2.8$ to $6$ at fixed $Ek=3\times10^{-7}$ for two aspect ratios and perform a $Ra$-scan at $Pr=3.7$, measuring $Nu$ and employing sidewall thermometry to assess wall modes. They find a significant $Pr$-dependence: increasing $Pr$ lowers $Nu$ by about $25\%$ across the tested range, and $Nu$ scales with $Ra$ more steeply under rotation than in the non-rotating case, with a collapse achieved by $Nu\propto Ra^{0.41}$. Wall modes are stronger at low $Pr$ and contribute to heat transfer, but bulk flow changes are also important. Overall, the results reveal strong $Pr$ sensitivity in geostrophic rotating convection and highlight the need to extend measurements and simulations to broader $Pr$ and geostrophic parameter regimes for geophysical relevance.

Abstract

Flows at planetary scales are generally driven by buoyancy and influenced by rotation. Rotating Rayleigh-Bénard convection (RRBC) is a practical and simple model that can be used to describe these systems. In RRBC, thermally induced convection occurs, which is influenced by the constant rotation it experiences. We study RRBC in a cylinder in the \red{transition region between rotation-affected and rotation-dominated (also called geostrophic) convection}. Experiments are performed to assess the dependence of the Nusselt number $\Nu$ (efficiency of convective heat transfer) on the Prandtl number $\Pr$ (ratio of kinematic viscosity over thermal diffusivity), a relation that is not explored much for geostrophic convection. By using water at different mean temperatures we can reach $2.8\le \Pr\le 6$. We study the relation between $\Pr$ and $\Nu$ at constant Ekman number $\Ek=3\times10^{-7}$ (an inverse measure for strength of rotation) for two different diameter-to-height aspect ratios ($Γ=1/5$ and $1/2$) of the setup. The corresponding constant Rayleigh numbers (strength of thermal forcing) are $\Ra=1.1\times 10^{12}$ and $1\times 10^{11}$, respectively. Additionally, we measure the relation between the Rayleigh number $\Ra$ and $\Nu$ for $4\times10^{10}\le \Ra\le 7\times10^{11}$, $\Ek=3\times10^{-7}$ and $\Pr=3.7$. It is found that $\Nu$ exhibits a significant dependence on $\Pr$, even within this limited range. Increasing $\Pr$ by a factor 2 resulted in a decrease of $\Nu$ of about $25 \%$. We hypothesize that the decrease of $\Nu$ is caused by the changing ratio of the thermal and kinetic boundary layer thicknesses as a result of increasing $\Pr$. We also consider the anticipated contributions of the wall mode to the heat transfer using sidewall temperature measurements.

Experiments on rapidly rotating convection: the role of the Prandtl number

TL;DR

This work investigates how heat transfer in rapidly rotating convection depends on the Prandtl number within the transition between rotation-affected and geostrophic convection. Using the TROCONVEX apparatus, the authors vary from to at fixed for two aspect ratios and perform a -scan at , measuring and employing sidewall thermometry to assess wall modes. They find a significant -dependence: increasing lowers by about across the tested range, and scales with more steeply under rotation than in the non-rotating case, with a collapse achieved by . Wall modes are stronger at low and contribute to heat transfer, but bulk flow changes are also important. Overall, the results reveal strong sensitivity in geostrophic rotating convection and highlight the need to extend measurements and simulations to broader and geostrophic parameter regimes for geophysical relevance.

Abstract

Flows at planetary scales are generally driven by buoyancy and influenced by rotation. Rotating Rayleigh-Bénard convection (RRBC) is a practical and simple model that can be used to describe these systems. In RRBC, thermally induced convection occurs, which is influenced by the constant rotation it experiences. We study RRBC in a cylinder in the \red{transition region between rotation-affected and rotation-dominated (also called geostrophic) convection}. Experiments are performed to assess the dependence of the Nusselt number (efficiency of convective heat transfer) on the Prandtl number (ratio of kinematic viscosity over thermal diffusivity), a relation that is not explored much for geostrophic convection. By using water at different mean temperatures we can reach . We study the relation between and at constant Ekman number (an inverse measure for strength of rotation) for two different diameter-to-height aspect ratios ( and ) of the setup. The corresponding constant Rayleigh numbers (strength of thermal forcing) are and , respectively. Additionally, we measure the relation between the Rayleigh number and for , and . It is found that exhibits a significant dependence on , even within this limited range. Increasing by a factor 2 resulted in a decrease of of about . We hypothesize that the decrease of is caused by the changing ratio of the thermal and kinetic boundary layer thicknesses as a result of increasing . We also consider the anticipated contributions of the wall mode to the heat transfer using sidewall temperature measurements.

Paper Structure

This paper contains 8 sections, 5 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Results and discussion
  4. Heat flux at constant $\mathrm{Ra}$
  5. Heat flux at constant $\mathrm{Pr}$
  6. Wall mode analysis
  7. Conclusion
  8. Experiment specifications

Figures (5)

  • Figure 1: Overview of conducted experiments. All our data points are at $\mathrm{Ek}=3\times 10^{-7}$. (a) $(\mathrm{Pr},\mathrm{Ra})$ diagram with regime boundaries from literature. Blue circles indicate the setup with $\Gamma=1/2$ while red squares indicate the setup with $\Gamma=1/5$. The measurements for $\Gamma=1/2$ were on a range of $\mathrm{Pr}$ at constant $\mathrm{Ra}=1\times 10^{11}$ as well as on a range of $\mathrm{Ra}$ at constant $\mathrm{Pr}=3.7$. The measurements for $\Gamma=1/5$ were on the same range of $\mathrm{Pr}$ but at $\mathrm{Ra}=1.1\times 10^{12}$. The transition between the nonrotating and rotation-affected range (green dashed line) is based on the aspect-ratio-dependent criterion (\ref{['eq:weiss2010']}) due to Weiss et al. Weiss2010, which at $\Gamma=1/5$ gives $\mathrm{Ro}=0.40$. Two suggested relations for the transition between the rotation-affected and geostrophic range are included: $\mathrm{Ra}=10\mathrm{Ek}^{-3/2}$ due to King et al. King2012King2013 (red dashed line) and $\mathrm{Ra}=\mathrm{Pr}^{3/5}\mathrm{Ek}^{-8/5}$ from Cheng et al. Cheng2020 (blue dashed line). The dotted line indicates $\mathrm{Ro}=0.1$ for reference. (b) $(\mathrm{Pr},\mathrm{Ra}/\mathrm{Ra}_c)$ diagram with a comparison of other literature data for rapidly rotating convection ($\mathrm{Ek}\le 10^{-6}$). Apart form our current data, we include the experimental works of Ecke & Niemela (2014) Ecke2014, Cheng et al. (2020) Cheng2020, Lu et al. (2021) Lu2021, Wedi et al. (2021) Wedi2021, Abbate & Aurnou (2023) Abbate2023; the cylinder DNS works of Favier & Knobloch (2020) Favier2020, de Wit et al. (2023) DeWit2023; and the periodic DNS works of Aguirre Guzmán et al. (2022) AguirreGuzman2022, Song et al. (2024) Song2024scalingSong2024. The corresponding $\widetilde{\mathrm{Ra}}=\mathrm{Ra}\mathrm{Ek}^{4/3}$ values are indicated (right ordinate).
  • Figure 2: (a) Plot of $\mathrm{Nu}$ as a function of $\mathrm{Pr}$ for $\Gamma=1/2$ (blue, $\mathrm{Ra} = 1\times10^{11}$, Ek$=3.1\times10^{-7}$, Ro=$0.040-0.056$) and $\Gamma=1/5$ (red, $\mathrm{Ra} = 1.1\times10^{12}$, Ek$=3\times10^{-7}$, Ro=$0.131-0.189$). (b) The same data compensated by the empirical factor $\mathrm{Ra}^{0.41}$. Note that in order to keep Ek constant, $\Omega$ was decreased as $\mathrm{Pr}$ decreased.
  • Figure 3: (a) Scaling of $\mathrm{Nu}$ as a function of $\mathrm{Ra}$. The blue markers indicate the current experimental data, with Pr$=3.7$ and Ek$=3\times10^{-7}$ ($\mathrm{Ro}=0.035-0.134$) and the purple dashed line indicating the fit of the relation $\mathrm{Nu}(\mathrm{Ra})$. The red markers indicate data from Cheng et al. Cheng2020, which is from experiments on the same setup but at higher $\mathrm{Pr}=5.2$ and at the same $\mathrm{Ek}=3\times10^{-7}$. Circles indicate $\Gamma=1/2$ and squares indicate $\Gamma=1/5$. The black line indicates the relation between $\mathrm{Nu}_{\textcolor{black}{0}}$ and $\mathrm{Ra}$ in non-rotating convection. (b) The same data, normalized with the nonrotating $\mathrm{Nu}_0$ result from Cheng et al. Cheng2020.
  • Figure 4: (a) Normalized magnitude $T_{rms}/\Delta T$ of temperature fluctuations as measured by the sidewall temperature probes. Digits $1,3,5$ indicate increasing distance from the bottom plate while labels A, B are for laterally opposite probes at the same height. (b) A schematic drawing of the setup with the sidewall sensors indicated in grey for both aspect ratios. The sensors were placed at midheight of each section of the cylinder.
  • Figure 5: Plot of relevant water properties as a function of temperature: thermal expansion coefficient $\alpha$, kinematic viscosity $\nu$, thermal diffusivity $\kappa$, thermal conductivity $k$ and the Prandtl number $\mathrm{Pr}=\nu/\kappa$. They are plotted relative to the reference temperature $T_{ref}=31^\circ$C of our previous study Cheng2020. Curves plotted according to polynomial relations given by Lide Lide2000. At $T_{ref}=31^\circ$C, these give $\alpha=3.15\times 10^{-4}$ K$^{-1}$, $\nu=7.73\times 10^{-7}$ m$^2$/s, $\kappa=1.48\times 10^{-7}$ m$^2$/s, $k=0.616$ W/(m K) and $\mathrm{Pr}=5.22$.