Maximal heat transfer between two parallel plates

Abstract

The divergence-free time-independent velocity vector field has been determined so as to maximise heat transfer between two parallel plates of a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh $etal$. (2014); however, a search range of optimal states has been extended to a three-dimensional velocity field. The scaling of the Nusselt number $Nu$ with the Péclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion timescale), $Nu\sim P{e}^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim R{a}^{1/2}$, in extremely-high-$Ra$ convective turbulence, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim 10^0$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at higher $Pe$. At $Pe\gtrsim 10^3$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic mean profile near the walls.

10 pages, 6 figures

References in corpus (5)

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• Logarithmic temperature profiles in turbulent Rayleigh-Bénard convection
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Cited by in corpus (8)

• Wall-to-wall optimal transport in two dimensions
• On the optimal design of wall-to-wall heat transport
• Coherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-Bénard Convection
• Identifying causal significance in three-dimensional isotropic turbulence
• Optimal cooling of an internally heated disc
• Logarithmic and nonlogarithmic scaling laws of two-point statistics in wall turbulence
• How vortices enhance heat transfer from an oscillating plate
• Enhancing wall-to-wall heat transport with unsteady flow perturbations