  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 $\it et{\ }al$. (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 P\'eclet number $Pe$ (i.e., the square root of the
non-dimensionalised enstrophy with thermal diffusion timescale), $Nu\sim
Pe^{2/3}$, has been found in the three-dimensional optimal states,
corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim
Ra^{1/2}$, in extremely-high-$Ra$ convective turbulence, and thus to the Taylor
energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim10^{0}$, 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 $Pe\gtrsim10^{3}$, 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.
