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1. Exact relations between Rayleigh–Bénard and rotating plane Couette flow in two dimensions

   https://doi.org/10.1017/jfm.2020.718  

   Eckhardt, Bruno ; Doering, Charles R. ; Whitehead, Jared P. ( November 2020 , Journal of Fluid Mechanics)

   null (Ed.)

   Rayleigh–Bénard convection (RBC) and Taylor–Couette flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial independent variables, there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using this, we deduce several relations between both flows: (i) heat and angular momentum transport differ by $(1-R_{\Omega })$ , explaining why angular momentum transport is not symmetric around $R_{\Omega }=1/2$ even though the relation between $Ra$ , the Rayleigh number, and $R_{\Omega }$ , a non-dimensional measure of the rotation, has this symmetry. This relationship leads to a predicted value of $R_{\Omega }$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full three-dimensional system. (ii) One variable in both flows satisfies a maximum principle, i.e. the fields’ extrema occur at the walls. Accordingly, backflow events in shear flow cannot occur in this quasi two-dimensional setting. (iii) For free-slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re_S^{5/6}$ , where $Re_S$ is the shear Reynolds number, with a scaling exponent smaller than the anticipated $Re_S^1$ . 

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2. Steady Rayleigh–Bénard convection between stress-free boundaries

   https://doi.org/10.1017/jfm.2020.812  

   Wen, Baole ; Goluskin, David ; LeDuc, Matthew ; Chini, Gregory P. ; Doering, Charles R. ( December 2020 , Journal of Fluid Mechanics)

   null (Ed.)

   Steady two-dimensional Rayleigh–Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios ${\rm \pi} /5\leqslant \varGamma \leqslant 4{\rm \pi}$ , where $\varGamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\leqslant Ra\leqslant 10^{11}$ , and four orders of magnitude in the Prandtl number, $10^{-2}\leqslant Pr\leqslant 10^2$ . At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$ . The prefactor of this scaling depends on $\varGamma$ and is largest at $\varGamma \approx 1.9$ . The Reynolds number $Re$ for large- $Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\varGamma \approx 4.5$ . All of these large- $Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox ( Phys. Fluids , vol. 21, 2009, 083603). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\varGamma$ is smaller. 

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3. Comparison of quasi-geostrophic, hybrid and 3-D models of planetary core convection

   https://doi.org/10.1093/gji/ggac141  

   Barrois, O. ; Gastine, T. ; Finlay, C. C. ( April 2022 , Geophysical Journal International)

   We present investigations of rapidly rotating convection in a thick spherical shell geometry relevant to planetary cores, comparing results from quasi-geostrophic (QG), 3-D and hybrid QG-3D models. The 170 reported calculations span Ekman numbers, Ek, between 10−4 and 10−10, Rayleigh numbers, Ra, between 2 and 150 times supercritical and Prandtl numbers, Pr, between 10 and 10−2. The default boundary conditions are no-slip at both the ICB and the CMB for the velocity field, with fixed temperatures at the ICB and the CMB. Cases driven by both homogeneous and inhomogeneous CMB heat flux patterns are also explored, the latter including lateral variations, as measured by Q*, the peak-to-peak amplitude of the pattern divided by its mean, taking values up to 5. The QG model is based on the open-source pizza code. We extend this in a hybrid approach to include the temperature field on a 3-D grid. In general, we find convection is dominated by zonal jets at mid-depths in the shell, with thermal Rossby waves prominent close to the outer boundary when the driving is weaker. For the thick spherical shell geometry studied here the hybrid method is best suited for studying convection at modest forcing, $Ra \le 10 \, Ra_c$ when Pr = 1, and departs from the 3-D model results at higher Ra, displaying systematically lower heat transport characterized by lower Nusselt and Reynolds numbers. We find that the lack of equatorially-antisymmetric motions and z-correlations between temperature and velocity in the buoyancy force contributes to the weaker flows in the hybrid formulation. On the other hand, the QG models yield broadly similar results to the 3-D models, for the specific aspect ratio and range of Rayleigh numbers explored here. We cannot point to major disagreements between these two data sets at Pr ≥ 0.1, with the QG model effectively more strongly driven than the hybrid case due to its cylindrically averaged thermal boundary conditions. When Pr is decreased, the range of agreement between the hybrid and 3-D models expands, for example up to $Ra \le 15 \, Ra_c$ at Pr = 0.1, indicating the hybrid method may be better suited to study convection in the low Pr regime. We thus observe a transition between two regimes: (i) at Pr ≥ 0.1 the QG and 3-D models agree in the studied range of Ra/Rac while the hybrid model fails when $Ra\gt 15\, Ra_c$ and (ii) at Pr = 0.01 the QG and 3-D models disagree for $Ra\gt 10\, Ra_c$ while the hybrid and 3-D models agree fairly well up to $Ra \sim 20\, Ra_c$. Models that include laterally varying heat flux at the outer boundary reproduce regional convection patterns that compare well with those found in similarly forced 3-D models. Previously proposed scaling laws for rapidly rotating convection are tested; our simulations are overall well described by a triple balance between Coriolis, inertia and Archimedean forces with the length-scale of the convection following the diffusion-free Rhines-scaling. The magnitude of Pr affects the number and the size of the jets with larger structures obtained at lower Pr. Higher velocities and lower heat transport are seen on decreasing Pr with the scaling behaviour of the convective velocity displaying a strong dependence on Pr. This study is an intermediate step towards a hybrid model of core convection also including 3-D magnetic effects.

    

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4. Effects of pore scale and conjugate heat transfer on thermal convection in porous media

   https://doi.org/10.1017/jfm.2022.491  

   Korba, David ; Li, Like ( August 2022 , Journal of Fluid Mechanics)

   The study of thermal convection in porous media is of both fundamental and practical interest. Typically, numerical studies have relied on the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, where convection dynamics are assumed to be controlled solely by the Rayleigh number ( Ra ). Nusselt numbers ( Nu ) from these models predict Nu – Ra scaling exponents of 0.9–0.95. However, experiments and direct numerical simulations (DNS) have suggested scaling exponents as low as 0.319. Recent findings for solutal convection between DNS and DOB models have demonstrated that the ‘pore-scale parameters’ not captured by the DOB equations greatly influence convection. Thermal convection also has the additional complication of different thermal transport properties (e.g. solid-to-fluid thermal conductivity ratio k s / k f and heat capacity ratio σ ) in different phases. Thus, in this work we compare results for thermal convection from the DNS and DOB equations. On the effects of pore size, DNS results show that Nu increases as pore size decreases. Mega-plumes are also found to be more frequent and smaller for reduced pore sizes. On the effects of conjugate heat transfer, two groups of cases (Group 1 with varying k s / k f at σ  = 1 and Group 2 with varying σ at k s / k f  = 1) are examined to compare the Nu – Ra relations at different porosity ( ϕ ) and k s / k f and σ values. Furthermore, we report that the boundary layer thickness is determined by the pore size in DNS results, while by both the Rayleigh number and the effective heat capacity ratio, $\bar{\phi } = \phi + (1 - \phi )\sigma$ , in the DOB model. 

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5. Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media

   https://doi.org/10.3390/fluids7120373  

   Liu, Chang ; Knobloch, Edgar ( December 2022 , Fluids)

   This work employs single-mode equations to study convection and double-diffusive convection in a porous medium where the Darcy law provides large-scale damping. We first consider thermal convection with salinity as a passive scalar. The single-mode solutions resembling steady convection rolls reproduce the qualitative behavior of root-mean-square and mean temperature profiles of time-dependent states at high Rayleigh numbers from direct numerical simulations (DNS). We also show that the single-mode solutions are consistent with the heat-exchanger model that describes well the mean temperature gradient in the interior. The Nusselt number predicted from the single-mode solutions exhibits a scaling law with Rayleigh number close to that followed by exact 2D steady convection rolls, although large aspect ratio DNS results indicate a faster increase. However, the single-mode solutions at a high wavenumber predict Nusselt numbers close to the DNS results in narrow domains. We also employ the single-mode equations to analyze the influence of active salinity, introducing a salinity contribution to the buoyancy, but with a smaller diffusivity than the temperature. The single-mode solutions are able to capture the stabilizing effect of an imposed salinity gradient and describe the standing and traveling wave behaviors observed in DNS. The Sherwood numbers obtained from single-mode solutions show a scaling law with the Lewis number that is close to the DNS computations with passive or active salinity. This work demonstrates that single-mode solutions can be successfully applied to this system whenever periodic or no-flux boundary conditions apply in the horizontal. 

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