Self-similar mixing in stratified plane Couette flow for varying Prandtl number

We investigate fully developed turbulence in statically stable stratified plane Couette flows (the flow between two horizontal plates a distance $2h$ apart moving at velocities $\pm U_0$ and held at densities $\rho_a \mp \rho_0$) using direct numerical simulations at a range of Prandtl numbers $Pr \equiv \nu/\kappa \in \{0.7,7,70\}$ and Reynolds numbers $Re\equiv U_0 h/\nu \in [865,280000]$. We observe significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile, which can be described through a mixing length model using Monin-Obukhov (M-O) similarity theory. We employ M-O theory to formulate similarity scalings for various flow diagnostics in the gap interior. The mid-channel-gap gradient Richardson number $Ri_g$ is determined by the length scale ratio $h/L$, where $L$ is the Obukhov length scale. When $h/L \gg 1$, $Ri_g$ asymptotes to a maximum characteristic value of approximately 0.2, for very high $Re$ and for a range of $Pr$ and bulk Richardson number $Ri=g \rho_0 h/(\rho_a U_0^2)$. The flux Richardson number $Ri_f \simeq Ri_g$, implying that such turbulent flows do not access the (strongly) `layered anisotropic stratified turbulence' regime, and that the turbulent Prandtl number is approximately one.