Rayleigh-Bénard convection: The container shape matters

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 $D$.

This article appears in the following collection:

Figure 1

Examples of eigenfunctions of the Laplace operator (possible dominating temperature modes close to the onset of convection in Rayleigh-Bénard convection) for two-dimensional domains and Dirichlet boundary conditions at the top ($z=H$) and bottom ($z=0$) boundaries and (a–c) Dirichlet boundary conditions at the side boundaries ($x=0$ and $x=D$) or (d–f) Neumann boundary conditions at the side boundaries, for different aspect ratios of the domain: (a, d) $\mathrm{\Gamma }=1$, (b, e) any case of $\mathrm{\Gamma }<1$ (shown example is for $\mathrm{\Gamma }=1/3$), and (c, f) $\mathrm{\Gamma }>1$ (shown example is for $\mathrm{\Gamma }=5$). All modes have a form (a–c) $\sim sin(\pi n_{x}x/D)sin(\pi n_{z}z/H)$ for Dirichlet boundary conditions at the side boundaries or (d–f) $\sim cos(\pi n_{x}x/D)sin(\pi n_{z}z/H)$ for Neumann boundary conditions at the side boundaries. Pink and blue correspond to warm or cold fluid, respectively.

Figure 2

Examples of eigenfunctions of the Laplace operator in a circle for (a–c) Dirichlet boundary conditions at the side boundary ($r=D/2$) or (d–f) Neumann boundary conditions at $r=D/2$. All shown modes have a form $\sim J_n(2{\alpha }_{nk}(r/D))cos\left(n\varphi \right)$, where ${\alpha }_{nk}$ is (a–c) the $k\mathrm{th}$ positive root $j_{nk}$ of the Bessel function $J_n$, for Dirichlet boundary conditions at at $r=D/2$, or (d–f) the $k\mathrm{th}$ positive root ${\tilde{j}}_{nk}$ of the derivative $J_n^{\prime }$ of the Bessel function $J_n$, for Neumann boundary conditions at at $r=D/2$. These can be interpreted as possible dominating temperature modes in the central horizontal cross section of a cylindrical Rayleigh-Bénard cell, close to the onset of convection, for Dirichlet boundary conditions at the top ($z=H$) and bottom ($z=0$) plates, where the reduced temperature $\theta$ vanishes, and (a–c) Dirichlet (conducting) boundary conditions at the sidewall ($r=D/2$) or (d–f) Neumann (adiabatic) boundary conditions at the sidewall ($r=D/2$), for different aspect ratios of the container: (a, d) $\mathrm{\Gamma }\le 1$, (b, e) $\mathrm{\Gamma }\approx 2$, and (c, f) $\mathrm{\Gamma }\approx 5$. Color scale as in Fig. 1.

Figure 3

Example of a pair of eigenfunctions, (a, c) “$2\times 2$ mode” and (b, d) “wall mode,” that correspond to the same (degenerate) eigenvalue of the Laplace operator (possible dominating temperature modes close to the onset of Rayleigh-Bénard convection) for a cylindrical domain with the aspect ratio $\mathrm{\Gamma }\approx 3.878$ and Dirichlet boundary conditions at the top ($z=H$) and bottom ($z=0$) boundaries and Neumann boundary conditions at the side boundary, which are plotted in the horizontal cross sections (a, b), for (a) $z/H=1/4$ and (b) $z/H=1/2$, and in the central vertical cross sections (c, d). Both eigenfunctions, in (a, c) and in (b, d), have a form $J_n(2{\tilde{j}}_{nk}r/D)cos\left(n\varphi \right)sin(\pi n_{z}z/H)$ and correspond to the same eigenvalue $\lambda =4{\tilde{j}}_{nk}^2/D^2+{\pi }^2{n}_z^2/H^2$, where (a, c) $n=1$, $k=1$, $n_z=2$ and $\lambda \approx 4{\pi }^2{H}^{-2}(1+0.34{\mathrm{\Gamma }}^{-2})$ and (b, d) $n=9$, $k=1$, $n_z=1$ and $\lambda \approx 4{\pi }^2{H}^{-2}(0.25+11.63{\mathrm{\Gamma }}^{-2})$, both for $\mathrm{\Gamma }\approx 3.878$. Here $J_n$ is the Bessel function $J_n$ and ${\tilde{j}}_{nk}$ is the $k\mathrm{th}$ root of the derivative of the Bessel function $J_n$. Pink and blue correspond to warm or cold fluid, respectively.

Figure 4

Another example of a pair of eigenfunctions, (a, c) “$2\times 2$ mode” and (b, d) “donut mode,” that correspond to the same (degenerate) eigenvalue of the Laplace operator (possible dominating temperature modes close to the onset of Rayleigh-Bénard convection) for a cylindrical domain with the aspect ratio $\mathrm{\Gamma }\approx 1.235$ and Dirichlet boundary conditions at the top ($z=H$) and bottom ($z=0$) boundaries and Neumann boundary conditions at the side boundary, which are plotted in the horizontal cross sections (a, b), for (a) $z/H=1/4$ and (b) $z/H=1/2$, and in the central vertical cross sections (c, d). Both eigenfunctions, in (a, c) and in (b, d), have a form $J_n(2{\tilde{j}}_{nk}r/D)cos\left(n\varphi \right)sin(\pi n_{z}z/H)$ and correspond to the same eigenvalue $\lambda =4{\tilde{j}}_{nk}^2/D^2+{\pi }^2{n}_z^2/H^2$, where (a, c) $n=1$, $k=1$, $n_z=2$, and $\lambda \approx 4{\pi }^2{H}^{-2}(1+0.34{\mathrm{\Gamma }}^{-2})$ and (b, d) $n=0$, $k=1$, $n_z=1$, and $\lambda \approx 4{\pi }^2{H}^{-2}(0.25+1.488{\mathrm{\Gamma }}^{-2})$, for $\mathrm{\Gamma }\approx 1.235$. Here $J_n$ is the Bessel function $J_n$ and ${\tilde{j}}_{nk}$ is the $k\mathrm{th}$ root of the derivative of the Bessel function $J_n$. Pink and blue correspond to warm or cold fluid, respectively.

Figure 5

Contours of the temperature fields in Rayleigh-Bénard convection close to the onset of convection, in parallelepiped containers with ${\mathrm{\Gamma }}_x={\mathrm{\Gamma }}_y$ (upper row) and cylindrical containers (lower row) with adiabatic sidewalls, for different aspect ratios $\mathrm{\Gamma }$: (a, h) $\mathrm{\Gamma }=1/10$, (b, i) $\mathrm{\Gamma }=1/5$, (c, j) $\mathrm{\Gamma }=1/2$, (d, k) $\mathrm{\Gamma }=1$, (e, l) $\mathrm{\Gamma }=2$, (f, m) $\mathrm{\Gamma }=5$, and (g, n) $\mathrm{\Gamma }=10$, as obtained numerically from the linearized governing equations. The color scale ranges from blue (cold fluid) to pink (warm fluid). Simulations conducted by P. Reiter; adopted from [59]. The flow structures in the horizontal direction are similar to the theoretical ones; cf. (h–k) with Fig. 2 for $\mathrm{\Gamma }\le 1$, (l) with Fig. 2 for $\mathrm{\Gamma }\approx 2$, and (m) with Fig. 2 for $\mathrm{\Gamma }\approx 5$.

Figure 6

Critical Rayleigh numbers ${\text{Ra}}_{c,\mathrm{\Gamma }}$ for the onset of convection, as functions of the domain aspect ratio $\mathrm{\Gamma }$: Linear growth rates (shown with colored vertically elongated boxes), obtained from numerical solutions of the linearized equations for fluctuations, initialized with random perturbations, and (a, b) the neutral stability curves (blue solid lines) as obtained from the classical linear stability analysis using the full spectrum of the normal modes, everywhere for no-slip boundary conditions for the velocity and (a, b) two-dimensional rectangular domains and (c, d) three-dimensional cylindrical domains, with (a, c) conducting sidewalls and (b, d) adiabatic sidewalls; adapted from [59]. Black dashed lines show fits of the numerical solutions of the linearized equations, sought in the form ${\text{Ra}}_{c,\mathrm{\Gamma }}\approx 1708{(1+C{\mathrm{\Gamma }}^{-2})}^2$, with the best-fit constants $C$ shown in the corresponding legends. Pink continuous lines show the theoretical estimates that we have derived in this paper; see Eq. (54) for (a), Eq. (55) for (b), Eq. (67) for (c), and Eq. (68) for (d). Our theoretical estimates (pink solid lines) and the linear stability results obtained numerically from the linearized equations (black dashed lines) show excellent agreement. Blue dashed lines show a theoretical estimate ${\text{Ra}}_{c,\infty }^J\approx 1708$ [23, 24] for a laterally unbounded fluid layer, obtained from the linear stability analysis of the normal modes, restricted to the consideration of a single mode in the horizontal direction.

Figure 7

(a) Solid lines: ratios of the critical Rayleigh numbers ${\text{Ra}}_{c,\mathrm{\Gamma }}$ for the onset of convection, as estimated in this study, and ${\text{Ra}}_{c,\mathrm{\Gamma }}$ as obtained from numerical solutions of the linearized equations for fluctuations (numerical linear stability analysis) [59], plotted versus the container aspect ratio $\mathrm{\Gamma }$. The numerical linear stability data are represented by their fits sought in a form ${\text{Ra}}_{c,\mathrm{\Gamma }}^{\text{LSA}}\approx 1708{(1+C/{\mathrm{\Gamma }}^2)}^2$, with the best-fit constants $C$ shown in the legends here and in Fig. 6. ${\text{Ra}}_{c,\mathrm{\Gamma }}$ of this study are taken from Eq. (54) for two-dimensional rectangular domains with conducting sidewalls, Eq. (55) for two-dimensional rectangular domains with adiabatic sidewalls, Eq. (67) for three-dimensional cylindrical domains with conducting sidewalls, and Eq. (68) for three-dimensional cylindrical domains with adiabatic sidewall. Everywhere no-slip boundary conditions for the velocity are considered. Dashed lines show similar data as the continuous ones but with a 5% correction of the fitting coefficients in the numerical linear stability data. Panel (b) is as (a), but the numerical linear stability data are represented by their fits sought in a form ${\text{Ra}}_{c,\mathrm{\Gamma }}^{\text{LSA}}\approx {\left(2\pi \right)}^4{(1+C/{\mathrm{\Gamma }}^2)}^2$, with the best-fit constants $C$ shown in the legend. Taking into account a very broad range of the considered aspect ratios $\mathrm{\Gamma }$ and a certain inaccuracy of the numerical linear stability analysis data and of their fits, one can conclude that our theoretical estimates and the numerically obtained linear stability data [59] are in good agreement.

Figure 8

Ratios of the critical Rayleigh numbers ${\text{Ra}}_{c,\mathrm{\Gamma }}$ for the onset of convection, as obtained from numerical solutions of the linearized equations for fluctuations (numerical linear stability analysis) [59], and ${\text{Ra}}_{c,\mathrm{\Gamma }}$ as estimated in this study, plotted versus the container aspect ratio $\mathrm{\Gamma }$. The numerical linear stability data are shown as is, without any fitting procedure. The values of ${\text{Ra}}_{c,\mathrm{\Gamma }}$ of this study are taken from Eq. (54) for two-dimensional rectangular domains with conducting sidewalls, Eq. (55) for two-dimensional rectangular domains with adiabatic sidewalls, Eq. (67) for three-dimensional cylindrical domains with conducting sidewalls, and Eq. (68) for three-dimensional cylindrical domains with adiabatic sidewall. Everywhere no-slip boundary conditions for the velocity are considered. Taking into account that within the considered $\mathrm{\Gamma }$-range, the values of ${\text{Ra}}_{c,\mathrm{\Gamma }}$ vary within 5 orders of magnitude, one can agree that our theoretical estimates and the numerically obtained linear stability data [59] are in good agreement.

Figure 9

Mixed modes, according to Eq. (95), which are combinations of the donut mode and wall mode that have similar eigenvalues. The contributions of the donut mode and wall mode are reflected in the prefactors ${\xi }_d$ and ${\xi }_w$, respectively.

Figure 10

Mixed modes, according to Eq. (96), which are combinations of three modes that correspond to similar eigenvalues. The contributions of the donut mode and wall mode are reflected in the prefactors ${\xi }_d$ and ${\xi }_w$, respectively.

Figure 11

Ratio of the critical Rayleigh numbers ${\text{Ra}}_{c,\mathrm{\Gamma }}$ for the onset of convection in a cylindrical container and in an optimal parallelepiped container with equal widths of the sidewalls (${\mathrm{\Gamma }}_y={\mathrm{\Gamma }}_x$), for the cases of the same aspect ratio $\mathrm{\Gamma }$ of the parallelepiped and cylindrical containers (blue line) or for the same area of the plates in the parallelepiped and cylindrical containers (pink line), as functions of the aspect ratio of the cylindrical container $\mathrm{\Gamma }$. Everywhere, the boundary conditions are no-slip for the velocity at all walls and isothermal at the plates, and adiabatic at the sidewalls for the temperature. Calculations are according to Eqs. (101, 102, 103).

Figure 12

(a) Ratios of the critical Rayleigh number, ${\text{Ra}}_{c,\mathrm{\Gamma }}({\mathrm{\Gamma }}_y/{\mathrm{\Gamma }}_x)$, for the onset of convection in a parallelepiped container with different widths of the sidewalls (${\mathrm{\Gamma }}_y\le {\mathrm{\Gamma }}_x$) and the critical Rayleigh number for the onset of convection in a similar container, ${\text{Ra}}_{c,\mathrm{\Gamma }}\left(1\right)$, for equal widths of the sidewalls (${\mathrm{\Gamma }}_y={\mathrm{\Gamma }}_x$), as functions of ${\mathrm{\Gamma }}_y/{\mathrm{\Gamma }}_x$. Shown are curves for ${\mathrm{\Gamma }}_x=0.1$, 0.5, 1, 2, and 10. (b) Ratios of the critical Rayleigh number ${\text{Ra}}_{c,\mathrm{\Gamma }}$ for the onset of convection in a parallelepiped container with different widths of the sidewalls (${\mathrm{\Gamma }}_y\le {\mathrm{\Gamma }}_x$) and the critical Rayleigh number for the onset of convection in a similar container with equal widths of the sidewalls (${\mathrm{\Gamma }}_y={\mathrm{\Gamma }}_x$), as functions of ${\mathrm{\Gamma }}_x=\mathrm{\Gamma }$. Shown are curves for ${\mathrm{\Gamma }}_y/{\mathrm{\Gamma }}_x=0.01$, 0.1, 0.2, and 0.5. Everywhere, the boundary conditions are no-slip for the velocity at all walls and isothermal at the plates, and adiabatic at the sidewalls for the temperature. In both panels, the calculations are according to Eq. (56).

Figure 13

The dependence of orders of magnitude in Rayleigh numbers that one can attain in Rayleigh-Bénard experiments in a cylindrical container with the adiabatic sidewall, as function of the cylinder aspect ratio $\mathrm{\Gamma }$. The maximal value, $m_{max}$, depends on ${\text{Ra}}_D=\alpha g{\mathrm{\Delta }}_{max}{D}^3/\left(\kappa \nu \right)$, i.e., on the maximal Rayleigh number, which is calculated based on the diameter of the plates, $D$, the maximal possible temperature difference between the plates that one can attain in the experiment, ${\mathrm{\Delta }}_{max}$, and the specific properties of the working fluid.

Figure 14

(a) Compensated $\text{Nu}$ vs $\text{Ra}$, as obtained in (almost) Oberbeck-Boussinesq experiments and numerical simulations of Rayleigh-Bénard convection in a cylinder or in a three-dimensional cell with periodic boundary conditions, for Prandtl numbers $\text{Pr}$ that correspond to water ($\text{Pr}\approx 4$) and air ($\text{Pr}\approx 1$), for different $\mathrm{\Gamma }$ [59, 87, 89, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139]. Most data are for $\mathrm{\Gamma }=1$ and 1/2, which form the shape of this dependence. The data for extremely small $\mathrm{\Gamma }$ ($\text{Pr}=4.38$) show no discernible dependence. Vertical lines indicate the onset of the transition at high $\text{Ra}$, observed in Göttingen experiments [126, 127, 128] (the onset moves to higher $\text{Ra}$ with decreasing $\mathrm{\Gamma }$). (b) Compensated $\text{Nu}$ vs $\text{Ra}$ based on the proper length scale $\ell$, for the same data as in (a). Now the data for extremely small $\mathrm{\Gamma }$ follow the general trend and the transition happens at the same location for all $\mathrm{\Gamma }$ (the vertical lines from (a) merge into one line). The legend on the right corresponds to both panels, (a) and (b). Figure adapted from [59].

Figure 15

Snapshots of the temperature for the three statistically stable turbulent states in two-dimensional turbulent Rayleigh-Bénard convection with no-slip boundary conditions at the top and bottom plates and periodic boundary conditions at the sides, for $\text{Ra}={10}^{10}$, $\text{Pr}=10$, $\mathrm{\Gamma }=8$: (a) ${\mathrm{\Gamma }}_r=2/3$ ($m=12$, $\text{Nu}\approx 105.2$), (b) ${\mathrm{\Gamma }}_r=4/5$ ($m=10$, $\text{Nu}\approx 101.5$), and (c) ${\mathrm{\Gamma }}_r=1$ ($m=8$, $\text{Nu}\approx 96.8$). One can see that for exactly the same control parameters $\text{Ra}$, $\text{Pr}$ and $\mathrm{\Gamma }$, the system can have different statistically steady states, with different number of rolls and different heat transport properties. Adopted from [153].