Optimal heat transport solutions for Rayleigh–Bénard convection

Abstract

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh–Bénard convection with no-slip horizontal walls for a variety of Prandtl numbers $\mathit{Pr}$ and Rayleigh number up to $\mathit{Ra}\sim 10^{9}$. Power-law scalings of $\mathit{Nu}\sim \mathit{Ra}^{{\it\gamma}}$ are observed with ${\it\gamma}\approx 0.31$, where the Nusselt number $\mathit{Nu}$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $\mathit{Pr}$ is found to be extremely weak. On the other hand, the presence of two local maxima of $\mathit{Nu}$ with different horizontal wavenumbers at the same $\mathit{Ra}$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $\mathit{Pr}\lesssim 7$, optimal transport is achieved at the smaller maximal wavenumber. In these fluids, the optimal structure is a plume of warm rising fluid, which spawns left/right horizontal arms near the top of the channel, leading to downdraughts adjacent to the central updraught. For $\mathit{Pr}>7$ at high enough $\mathit{Ra}$, the optimal structure is a single updraught lacking significant horizontal structure, and characterized by the larger maximal wavenumber.

Rayleigh–Bénard convection with a melting boundary

“…The presence of the topography induced by the convective flow itself seems to favor stable quasi-steady rolls as opposed to oscillatory ones. This leads to an increase in heat flux when compared to classical RB and is closer to the optimal solution of Sondak et al (2015), derived assuming steady laminar solutions. This marginal increase in the Nusselt number was also recently reported in the independent study by Rabbanipour Esfahani et al (2018), both in two and three dimensions, although for a Prandtl number of 10.…”

“…For reference, we also show some typical values obtained for classical Rayleigh number (each point corresponding in that case to the time average of a single simulation at fixed Rayleigh number). We also indicate the results of Sondak et al (2015) which correspond to the optimal heat transfer for a 2D steady solution giving N u ≈ 0.125Ra 0.31 †. Interestingly, although our simulation departs significantly from classical Rayleigh-Bénard, our renormalization shows that the Nusselt number follows that of classical RB convection in a quasi-static manner.…”

“…† InSondak et al (2015), the best fit is actually N u ≈ 0.115Ra 0.31 but corresponds to a Prandtl number of 7. They obtained a slightly larger prefactor for σ = 1 but the same power law.…”

Roughness as a Route to the Ultimate Regime of Thermal Convection

“…An important aspect emerging from the study of planar Rayleigh-Bénard convection in two dimensions for P r ≥ 1 is that the flow field [27] and the N u-Ra scaling relations [25,26,28] are similar to those in three dimensions. Thus, this correspondence permits one to understand the processes driving the heat transport using well resolved two-dimensional simulations.…”

Optimal Heat Transfer and Optimal Exit Times

“…Much of the current progress in designing industrial heat exchanger devices relies on direct numerical simulations in complex flows and geometries [16]. In a more theoretical context, flow patterns have been optimized to improve Rayleigh-Bénard convection [21,28] or to achieve maximal heat transport in simple 2D geometries [8,25,2,1]. The optimal distribution of sources and sinks has also been investigated for optimal transport of a passive scalar, whether heat or tracer [20,24,22].…”