New variational bounds on convective transport. II. Computations and implications

We study the maximal rate of scalar transport between parallel walls separated by distance $h$, by an incompressible fluid with scalar diffusion coefficient $\kappa$. Given velocity vector field ${\bf u}$ with intensity measured by the P\'eclet number $Pe = h^2 \langle | \nabla {\bf u} |^2 \rangle^{1/2}/\kappa$ (where $\langle\cdot\rangle$ is space-time average) the challenge is to determine the largest enhancement of wall-to-wall scalar flux over purely diffusive transport, i.e., the Nusselt number $Nu$. Variational formulations of the problem are studied numerically and optimizing flow fields are computed over a range of $Pe$. Implications of this optimal wall-to-wall transport problem for the classical problem of Rayleigh-B\'enard convection are discussed: the maximal scaling $Nu \sim Pe^{2/3}$ corresponds, via the identity $Pe^2 = Ra(Nu-1)$ where $Ra$ is the usual Rayleigh number, to $Nu \sim Ra^{1/2}$ as $Ra \rightarrow \infty$.