  We discuss what is an optimal velocity field for more heat transfer and less
energy dissipation under the constraints of the continuity equation for the
velocity and the advection-diffusion equation for temperature in plane Couette
flow. The excess of a wall heat flux (or equivalently total scalar dissipation)
over total energy dissipation is taken as an objective functional, and by using
a variational method the Euler-Lagrange equations are derived, which are solved
numerically to obtain the optimal states in the sense of maximisation of the
functional. At high Reynolds numbers, the optimal heat transfer is found in
three-dimensional velocity field in which hierarchical self-similar
quasi-streamwise vortical structures appear. The streamwise vortices are tilted
in the spanwise direction so that they may produce the anticyclonic vorticity
antiparallel to the mean-shear vorticity, bringing about significant
three-dimensionality. The isotherms wrapped around the tilted anticyclonic
vortices undergo the cross-axial shear of the mean flow, so that the spacing of
the wrapped isotherms is narrower and so the temperature gradient is steeper
than those around a purely streamwise (two-dimensional) vortex tube,
intensifying scalar dissipation and so a wall heat flux. Moreover, the tilted
anticyclonic vortices induce the flow towards the wall to push low- (or high-)
temperature fluids on the hot (or cold) wall, enhancing a wall heat flux. The
optimised three-dimensional velocity fields achieve a much higher wall heat
flux and much lower energy dissipation than those of plane Couette turbulence.
