Optimal heat transfer enhancement in plane Couette flow

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.