Maximal mixing rate in turbulent stably stratified Couette flow

A rigorous upper bound on the long-time-averaged vertical buoyancy flux is derived from the Navier–Stokes equations for a Boussinesq fluid confined between two parallel horizontal plates a distance d apart, maintained at a constant statically stabilizing temperature difference ΔT and driven at a constant relative velocity ΔU. The upper bound on the volume and long-time-averaged vertical buoyancy flux B≔limt→∞1/t∫0t 〈ρu3〉g/ρ0 dt̃ is B⩽Bmax=(1−16√/Re)(ΔU)3/(64√d), where Re=ΔUd/ν and ρ0 is some reference density. Significantly, Bmax is independent of the bulk Richardson number of the flow and is achieved by an optimal solution with a mixing efficiency (or flux Richardson number) which approaches 0.5 as the Reynolds number becomes large. The time-averaged turbulent dissipation of kinetic energy and the time-averaged vertical buoyancy flux are then in equipartition for the optimizing flow.