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 DeltaT and driven at a constant relative velocity DeltaU. The upper bound on the volume and long-time-averaged vertical buoyancy flux B:=lim(t --> infinity)1/t integral (t)(0) < rhou(3)>g/rho (0) d (t) over tilde is B less than or equal toB(max)=(1-16 root2>/Re)(DeltaU)(3)/(64 root2>d), where Re=Delta Ud/nu and rho (0) is some reference density. Significantly, B-max 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. (C) 2001 American Institute of Physics.