Wall to wall optimal transport

The calculus of variations is employed to find steady divergence-free velocity fields that maximize transport of a tracer between two parallel walls held at fixed concentration for one of two constraints on flow strength: a fixed value of the kinetic energy (mean square velocity) or a fixed value of the enstrophy (mean square vorticity). The optimizing flows consist of an array of (convection) cells of a particular aspect ratio $ #1{} #1{ {#1}} ≤ = ≤ = ≥ = ≥ = {Pr} {Fr} {Re} $ . We solve the nonlinear Euler–Lagrange equations analytically for weak flows and numerically – as well as via matched asymptotic analysis in the fixed energy case – for strong flows. We report the results in terms of the Nusselt number Nu , a dimensionless measure of the tracer transport, as a function of the Péclet number Pe , a dimensionless measure of the strength of the flow. For both constraints, the maximum transport NuMAX(Pe) is realized in cells of decreasing aspect ratio ₒₚₜ(Pe) as Pe increases. For the fixed energy problem, NuMAX ~ Pe and ₒₚₜ ~ Pe-1/2 , while for the fixed enstrophy scenario, NuMAX ~ Pe10/17 and ₒₚₜ ~ Pe-0.36 . We interpret our results in the context of buoyancy-driven Rayleigh–Bénard convection problems that satisfy the flow intensity constraints, enabling us to investigate how the transport scalings compare with upper bounds on Nu expressed as a function of the Rayleigh number Ra . For steady convection in porous media, corresponding to the fixed energy problem, we find NuMAX ~ Ra and ₒₚₜ ~ Ra-1/2 , while for steady convection in a pure fluid layer between stress-free isothermal walls, corresponding to fixed enstrophy transport, NuMAX ~ Ra5/12 and ₒₚₜ ~ Ra-1/4 .