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Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh–Bénard convection (RBC) exhibits the classical 1 / 3 scaling as the Rayleigh number R a → ∞ with Prandtl number unity, consistent with Malkus–Howard’s marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin–Doering–Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same 5 / 12 scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the 1 / 3 scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’. 

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. 

Abstract Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm—a negative Sobolev norm—and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q -recurrent ; otherwise it is q - transient . We use this classification to study several examples and raise questions for future investigations. 

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. 

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. 

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 $$Re_S^1$$ .