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#### Rayleigh-Bénard convection: The container shape matters

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##### Citation

Shishkina, O. (2021). Rayleigh-Bénard convection: The container shape matters.* Physical Review Fluids,* *6*: 090502, pp. 1. doi:10.1103/PhysRevFluids.6.090502.

Cite as: https://hdl.handle.net/21.11116/0000-000A-39C4-9

##### Abstract

To study turbulent thermal convection, one often chooses a Rayleigh-Benard 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-Benard 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, Delta(max), one ponders: Which shape of the plates and aspect ratio 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, Ra-c,Ra- Gamma. First we prove that Ra-c,Ra- Gamma proportional to (1 + c(u) Gamma(-2))(1 + c(theta) 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 Ra-c,Ra- (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 Ra-c,Ra- Gamma approximate to (2 pi)(4)(1 + c(u)Gamma(-2))(1 + c(theta)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) approximate to 1.49 and c(theta) = ((j) over tilde (11))(2)/pi(2) approximate to 0.34, where j(11) and (j) over tilde (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 Gamma(x) and Gamma(y), Gamma(y) <= Gamma(x) equivalent to Gamma, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain Ra-c,Ra- Gamma approximate to (2 pi)(4)(1 + Gamma(-2)(x))(1 + Gamma(-2)(x)/4 + Gamma(-2)(y)/4). Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For Gamma less than or similar to 4.4, the derived Ra-c,Ra-Gamma is larger than Jeffreys' result Ra-c,infinity(J) approximate to 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 Gamma -> infinity, the difference between Ra-c,Ra-Gamma ->infinity = (2 pi)(4) for laterally confined containers and Jeffreys' Ra-c,infinity(J) for an unbounded layer is about 8.8%.

We further show that in Rayleigh-Benard 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 Gamma is independent of Delta(max) and of the fluid properties. For the adiabatic sidewalls, the optimal Gamma is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Gamma = 1/2 in most Rayleigh-Benard experiments is right and justified. For the given plate diameter D and maximal temperature difference 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 Delta(max). Deviations from the optimal Gamma lead to a reduction of the attainable range, namely, as log(10) (Gamma) for Gamma -> 0 and as log(10) (Gamma(-3)) for Gamma -> infinity. Our theory shows that the relevant length scale in Rayleigh-Benard convection in containers with no-slip boundaries is l similar to D/root Gamma(2) + c(u) = H/root 1 + c(u)/Gamma(2). This means that in the limit Gamma -> infinity, l equals the cell height H, while for Gamma -> 0, it is rather the plate diameter D.

We further show that in Rayleigh-Benard 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 Gamma is independent of Delta(max) and of the fluid properties. For the adiabatic sidewalls, the optimal Gamma is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Gamma = 1/2 in most Rayleigh-Benard experiments is right and justified. For the given plate diameter D and maximal temperature difference 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 Delta(max). Deviations from the optimal Gamma lead to a reduction of the attainable range, namely, as log(10) (Gamma) for Gamma -> 0 and as log(10) (Gamma(-3)) for Gamma -> infinity. Our theory shows that the relevant length scale in Rayleigh-Benard convection in containers with no-slip boundaries is l similar to D/root Gamma(2) + c(u) = H/root 1 + c(u)/Gamma(2). This means that in the limit Gamma -> infinity, l equals the cell height H, while for Gamma -> 0, it is rather the plate diameter D.