Transition to the ultimat 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 nonlinear-normal 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.

Figures (8)

• Figure 1: (a) Schematic illustration of the SID experiment. (b) Detail of the mid-duct region in the duct reference frame, showing typical vertical profiles of mean streamwise velocity (blue) and density (green).
• Figure 2: (a-f) Snapshots of the spanwise mid-plane density field $\rho$ for increasingly turbulent flow regimes. The flow directions are indicated by arrows in (a). The $z$-axis is scaled up by 1.5. Simulations include artificial restoring forces applied in the blue regions, see Supplemental Material. Statistical quantities are averaged within $x/H\in[-14,14]$ (black dashed lines) to eliminate edge effects. (g) Qualitative flow regime phase diagram illustrating the current DNS results alongside previous experimental lefauve_buoyancy-driven_2020 and numerical zhu_stratified_2023 data at $\mathrm{Pr}=7$.
• Figure 3: (a) Log-log plot of $\mathrm{Nu}$ against $\mathrm{Ra}$. The fill color represents the flow regime as indicated in \ref{['fig:flow']}g and the shape corresponds to the inclination. The slope for small $\mathrm{Ra}$ is consistent with $\mathrm{Nu}\sim \mathrm{Ra}^{1/3}$ (dashed line). Scaling in the ultimate regime is guided by the red line $\mathrm{Ra}^{1/2}$. (b) Same data as (a), but compensated by $\mathrm{Ra}^{1/3}$. The inset shows the data in the ultimate regime compensated by $\mathrm{Ra}^{1/2}$. (c) Hydraulic dimensionless mass flux $Q_m$ against $\mathrm{Ra}$, the vertical scale is linear. The dashed line marks the hydraulic limit $Q_m= 0.5$.
• Figure 4: (a) Mean streamwise density profiles in $x/H\in[-14,14]$ of $\theta=7^\circ$, satisfying $(\rho|_{x \approx \pm 20H} - \rho_0) / \Delta\rho = \pm 0.5$ in the reservoirs. Deeper shade of color indicates higher Ra. (b) Density gradient $\beta$ for all three angles. The dashed lines in (a,b) marks the asymptotic value of $\beta=0.01$ at large Ra. (c) Current DNS results (large filled markers, showing only data with positive $\beta$) compared with CVC data ($\theta=90^\circ$) Tovar2002tisserand_convection_2010schmidt_axially_2012riedinger_heat_2013rusaouen_echanges_2014rusaouen_echanges_2014pawar_two_2016. Previous data are from experiments using heat in water ($\mathrm{Pr} \approx 7$), unless otherwise indicated in the legend. For a direct comparison with our large inclination DNS results, see Supplemental Material.
• Figure 5: Log-log plot of shear Reynolds number $\mathrm{Re}_s$ against $\mathrm{Ra}$. The purple dotted line and strip marks $\mathrm{Re}_s^*=420$ around which the laminar BLs are expected to become turbulent, emphasizing the subcritical nature of the transition, which occurs over a range of $\mathrm{Re}_s$. The fitted slopes match the expected $\mathrm{Re}_s\sim\mathrm{Ra}^{1/4}$ (dashed) for laminar BL and $\mathrm{Re}_s\sim\mathrm{Ra}^{1/2}$ (red solid) for turbulent BL.