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 $\rho_0$, kinematic viscosity $\nu$, and thermal diffusivity $\kappa$) confined between two parallel horizontal plates separated by a distance $d$, which are driven at a constant relative velocity $\uDelta U$, and are maintained at a constant (statically stable) temperature difference leading to a constant density difference $\uDelta \rho$. 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. We find that the associated (dimensional) upper bound ${\cal B}^*_{\hbox{\scriptsize\it max}}$ on the long-time-averaged and volume averaged buoyancy flux ${\cal B}^*:=\lim_{t \rightarrow \infty} (1/t) \int^t_0 \langle \rho u_3 \rangle g/\rho_{0} {\rm d}\skew2\tilde{t}$ (where $u_3$ is the vertical velocity, $g$ is the acceleration due to gravity, and angled brackets denote volume averaging) does not depend on either the bulk Richardson number $J=g\uDelta \rho d/(\rho_0 \uDelta U^2)$ of the flow, or the Prandtl number $\sigma=\nu/\kappa$ of the fluid. We show that ${\cal B}^*_{\hbox{\scriptsize\it max}}$ has the same inertial characteristic scaling as the (dimensional) mechanical energy dissipation rate ${\cal E}^*_B$, and ${\cal B}^*_{\hbox{\scriptsize\it max}}=0.001267\uDelta U^3/d$ as $\hbox{\it Re}\rightarrow\infty$. The associated flow structure exhibits velocity boundary layers embedded within density boundary layers, with local gradient Richardson numbers $\hbox{\it Ri} =O(\sigma/\hbox{\it Re})\ll 1$ in the vicinity of the horizontal plates. There is a correspondence between the predicted flow structure and the flow structure at a lower Reynolds number associated with the upper bound on the mechanical energy dissipation rate ${\cal E}^*_{\hbox{\scriptsize\it max}}$ in an unstratified fluid. We establish that, for the flow that maximizes the buoyancy flux, the flux Richardson number $\hbox{\it Ri}_f \rightarrow 1/3$ as $\hbox{\it Re} \rightarrow \infty$, independently to leading order of both $\hbox{\it Re}$ and $J$. There is a generic partition of the energy input by the shear into the fluid into three equal parts: viscous dissipation of kinetic energy by the mean flow; viscous dissipation of kinetic energy by perturbation velocities; and vertical buoyancy flux.