To study turbulent thermal convection, one often chooses a Rayleigh-Bénard flow configuration, where a fluid is confined between a heated bottom plate, a cooled top plate of the same shape, and insulated vertical sidewalls. When designing a Rayleigh-Bénard setup, for specified fluid properties under Oberbeck-Boussinesq conditions, the maximal size of the plates (diameter or area), and maximal temperature difference between the plates, ${\mathrm{\Delta }}_{max}$, one ponders: Which shape of the plates and aspect ratio $\mathrm{\Gamma }$ of the container (ratio between its horizontal and vertical extensions) would be optimal? In this article, we aim to answer this question, where under the optimal container shape, we understand such a shape, which maximizes the range between the maximal accessible Rayleigh number and the critical Rayleigh number for the onset of convection in the considered setup, ${\text{Ra}}_{c,\mathrm{\Gamma }}$. First we prove that ${\text{Ra}}_{c,\mathrm{\Gamma }}\propto (1+c_u{\mathrm{\Gamma }}^{-2})(1+c_{\theta }{\mathrm{\Gamma }}^{-2})$, for some $c_u>0$ and $c_{\theta }>0$. This holds for all containers with no-slip boundaries, which have a shape of a right cylinder, whose bounding plates are convex domains, not necessarily circular. Furthermore, we derive accurate estimates of ${\text{Ra}}_{c,\mathrm{\Gamma }}$, under the assumption that in the expansions (in terms of the Laplace eigenfunctions) of the velocity and reduced temperature at the onset of convection, the contributions of the constant-sign eigenfunctions vanish, both in the vertical and at least in one horizontal direction. With that we derive ${\text{Ra}}_{c,\mathrm{\Gamma }}\approx {\left(2\pi \right)}^4(1+c_u{\mathrm{\Gamma }}^{-2})(1+c_{\theta }{\mathrm{\Gamma }}^{-2})$, where $c_u$ and $c_{\theta }$ are determined by the container shape and boundary conditions for the velocity and temperature, respectively. In particular, for circular cylindrical containers with no-slip and insulated sidewalls, we have $c_u=j_{11}^2/{\pi }^2\approx 1.49$ and $c_{\theta }={\left({\tilde{j}}_{11}\right)}^2/{\pi }^2\approx 0.34$, where $j_{11}$ and ${\tilde{j}}_{11}$ are the first positive roots of the Bessel function $J_1$ of the first kind or its derivative, respectively. For parallelepiped containers with the ratios ${\mathrm{\Gamma }}_x$ and ${\mathrm{\Gamma }}_y$, ${\mathrm{\Gamma }}_y\le {\mathrm{\Gamma }}_x\equiv \mathrm{\Gamma }$, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain ${\text{Ra}}_{c,\mathrm{\Gamma }}\approx {\left(2\pi \right)}^4(1+{\mathrm{\Gamma }}_x^{-2})(1+{\mathrm{\Gamma }}_x^{-2}/4+{\mathrm{\Gamma }}_y^{-2}/4)$. Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For $\mathrm{\Gamma }\lesssim 4.4$, the derived ${\text{Ra}}_{c,\mathrm{\Gamma }}$ is larger than Jeffreys' result ${\text{Ra}}_{c,\infty }^J\approx 1708$ for an unbounded layer, which was obtained with linear stability analysis of the normal modes restricted to the consideration of a single perturbation wave in the horizontal direction. In the limit $\mathrm{\Gamma }\to \infty$, the difference between ${\text{Ra}}_{c,\mathrm{\Gamma }\to \infty }={\left(2\pi \right)}^4$ for laterally confined containers and Jeffreys' ${\text{Ra}}_{c,\infty }^J$ for an unbounded layer is about 8.8%. We further show that in Rayleigh-Bénard experiments, the optimal rectangular plates are squares, while among all convex plane domains, circles seem to match the optimal shape of the plates. The optimal $\mathrm{\Gamma }$ is independent of ${\mathrm{\Delta }}_{max}$ and of the fluid properties. For the adiabatic sidewalls, the optimal $\mathrm{\Gamma }$ is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of $\mathrm{\Gamma }=1/2$ in most Rayleigh-Bénard experiments is right and justified. For the given plate diameter $D$ and maximal temperature difference ${\mathrm{\Delta }}_{max}$, the maximal attainable Rayleigh number range is about 3.5 orders of magnitudes smaller than the order of the Rayleigh number based on $D$ and ${\mathrm{\Delta }}_{max}$. Deviations from the optimal $\mathrm{\Gamma }$ lead to a reduction of the attainable range, namely, as ${log}_{10}\left(\mathrm{\Gamma }\right)$ for $\mathrm{\Gamma }\to 0$ and as ${log}_{10}\left({\mathrm{\Gamma }}^{-3}\right)$ for $\mathrm{\Gamma }\to \infty$. Our theory shows that the relevant length scale in Rayleigh-Bénard convection in containers with no-slip boundaries is $\ell \sim D/\sqrt{{\mathrm{\Gamma }}^2+c_u}=H/\sqrt{1+c_u/{\mathrm{\Gamma }}^2}$. This means that in the limit $\mathrm{\Gamma }\to \infty$, $\ell$ equals the cell height $H$, while for $\mathrm{\Gamma }\to 0$, it is rather the plate diameter $D$.

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• Rayleigh-Bénard convection