More Like this

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 $Re_S^1$ . 

   more » « less
2. Onset of thermal convection in non-colloidal suspensions

   https://doi.org/10.1017/jfm.2021.59  

   Kang, Changwoo ; Yoshikawa, Harunori N. ; Mirbod, Parisa ( May 2021 , Journal of Fluid Mechanics)

   null (Ed.)

   This study explores thermal convection in suspensions of neutrally buoyant, non-colloidal suspensions confined between horizontal plates. A constitutive diffusion equation is used to model the dynamics of the particles suspended in a viscous fluid and it is coupled with the flow equations. We employ a simple model that was proposed by Metzger, Rahli & Yin ( J. Fluid Mech. , vol. 724, 2013, pp. 527–552) for the effective thermal diffusivity of suspensions. This model considers the effect of shear-induced diffusion and gives the thermal diffusivity increasing linearly with the thermal Péclet number ( Pe ) and the particle volume fraction ( ϕ ). Both linear stability analysis and numerical simulation based on the mathematical models are performed for various bulk particle volume fractions $({\phi _b})$ ranging from 0 to 0.3. The critical Rayleigh number $(R{a_c})$ grows gradually by increasing ${\phi _b}$ from the critical value $(R{a_c} = 1708)$ for a pure Newtonian fluid, while the critical wavenumber $({k_c})$ remains constant at 3.12. The transition from the conduction state of suspensions is subcritical, whereas it is supercritical for the convection in a pure Newtonian fluid $({\phi _b} = 0)$ . The heat transfer in moderately dense suspensions $({\phi _b} = 0.2\text{--}0.3)$ is significantly enhanced by convection rolls for small Rayleigh number ( Ra ) close to $R{a_c}$ . We also found a power-law increase of the Nusselt number ( Nu ) with Ra , namely, $Nu\sim R{a^b}$ for relatively large values of Ra where the scaling exponent b decreases with ${\phi _b}$ . Finally, it turns out that the shear-induced migration of particles can modify the heat transfer. 

   more » « less
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.

    

   more » « less
4. Thermal convection over fractal surfaces

   https://doi.org/10.1017/jfm.2020.826  

   Toppaladoddi, Srikanth ; Wells, Andrew J. ; Doering, Charles R. ; Wettlaufer, John S. ( January 2021 , Journal of Fluid Mechanics)

   null (Ed.)

   We use well resolved numerical simulations with the lattice Boltzmann method to study Rayleigh–Bénard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $Ra \in \left [10^7, 10^{10}\right ]$ , where Pr and Ra are the Prandtl and Rayleigh numbers. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $k$ , as $S(k) \sim k^{p}$ ( $p < 0$ ). The degree of roughness is quantified by the exponent $p$ with $p < -3$ for smooth (differentiable) surfaces and $-3 \le p < -1$ for rough surfaces with Hausdorff dimension $D_f=\frac {1}{2}(p+5)$ . By computing the exponent $\beta$ using power law fits of $Nu \sim Ra^{\beta }$ , where $Nu$ is the Nusselt number, we find that the heat transport scaling increases with roughness through the top two decades of $Ra \in \left [10^8, 10^{10}\right ]$ . For $p$ $= -3.0$ , $-2.0$ and $-1.5$ we find $\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$ and $0.352 \pm 0.011$ , respectively. We also find that the Reynolds number, $Re$ , scales as $Re \sim Ra^{\xi }$ , where $\xi \approx 0.57$ over $Ra \in \left [10^7, 10^{10}\right ]$ , for all $p$ used in the study. For a given value of $p$ , the averaged $Nu$ and $Re$ are insensitive to the specific realization of the roughness. 

   more » « less
5. Steady Rayleigh–Bénard convection between no-slip boundaries

   https://doi.org/10.1017/jfm.2021.1042  

   Wen, Baole ; Goluskin, David ; Doering, Charles R. ( February 2022 , Journal of Fluid Mechanics)

   The central open question about Rayleigh–Bénard convection – buoyancy-driven flow in a fluid layer heated from below and cooled from above – is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed ‘classical’ $Nu \sim Ra^{1/3}$ or ‘ultimate’ $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$ , we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$ , it cannot achieve the ultimate scaling. 

   more » « less