Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified Couette flow

Abstract

We derive an improved rigorous upper bound for the long-time-averaged vertical buoyancy flux for stably stratified Couette flow; i.e. the flow of a Boussinesq fluid (with reference density p o , kinematic viscosity v, and thermal diffusivity K) confined between two parallel horizontal plates separated by a distance d, which are driven at a constant relative velocity ΔU, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference Δp. We construct the bound by means of a numerical solution to the background method' variation problem as formulated by Constantin and Doering using a one-dimensional uni-directional background. The upper bound so constructed is the best possible bound with the imposed constraints for streamwise independent mean flows that are statistically steady, and is calculated up to asymptotically large Reynolds numbers