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. 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 Peclet number Pe, a dimensionless measure of the strength of the flow. For both constraints, the maximum transport Nu(MAX). (Pe) is realized in cells of decreasing aspect ratio opt. Pe / as Pe increases. For the fixed energy problem, Nu(MAX) similar to Pe and opt similar to Pe 1 = 2, while for the fixed enstrophy scenario, Nu(MAX) similar to Pe10 = 17 and opt similar to Pe 0 : 36. We interpret our results in the context of buoyancy- driven Rayleigh- Benard 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 Nu(MAX) similar to Ra and opt similar to Ra 1 = 2, while for steady convection in a pure fluid layer between stress- free isothermal walls, corresponding to fixed enstrophy transport, Nu(MAX) similar to Ra5 = 12 and opt similar to Ra 1 = 4.