Transition to the ultimate regime of turbulent convection in stratified inclined duct flow

Abstract

The stratified inclined duct (SID) flow provides a canonical model for sustained, buoyancy-driven exchange between two reservoirs of different density, and emerges as a new paradigm in geophysical fluid dynamics. Yet, the flow dynamics remain unclear in the highly turbulent regime; laboratory experiments can access this regime but they lack resolution, while direct numerical simulations (DNS) at realistically high Prandtl number $\mathrm{Pr}=7$ (for heat in water) have not achieved sufficiently high Reynolds numbers $\mathrm{Re}$. We conduct three-dimensional DNS up to $\mathrm{Re}= 8000$ and observe the transition to the so-called ultimate regime of turbulent convection as evidenced by the Nusselt number scaling $\mathrm{Nu} \sim \mathrm{Ra}^{1/2}$, i.e., considerably enhanced transport. At the transition the shear Reynolds number, a key parameter characterizing boundary layer (BL) dynamics, exceeds the threshold range of 420 for turbulent kinetic BLs with the emergence of logarithmic velocity profiles. The nature of the transition towards ultimate SID flow is of non-normal-nonlinear nature, i.e., subcritical and hysteretic, as typical for the transition to fully turbulent shear flows. Our work connects SID flow with the broader class of wall-bounded turbulent convection flows and gives insight into mixing in the vigorously turbulent regimes in oceanography and industry.