Changes in the boundary-layer structure at the edge of the ultimate regime in vertical natural convection

Abstract

In thermal convection for very large Rayleigh numbers ($Ra$), the thermal and viscous boundary layers (BL) undergo a transition from a classical state to an ultimate state. In the former state, the BL thicknesses follow a laminar-like Prandtl-Blasius-Polhausen scaling, whereas in the latter, the BLs are turbulent with log-corrections in the sense of Prandtl and von K\'arm\'an. Here, we report evidence of this transition via changes in the BL structure of vertical natural convection (VC), which is a buoyancy driven flow between differentially heated vertical walls. The dataset spans $Ra$-values from $10^5$ to $10^9$ and Prandtl number value of 0.709. For this $Ra$ range, the VC flow exhibits classical state behaviour in a global sense. Yet, with increasing $Ra$, we observe that near-wall higher-shear patches occupy increasingly larger fractions of the wall-areas, which suggest that the BLs are undergoing a transition from the classical state to the ultimate shear-dominated state. The presence of streaky structures-reminiscent of the near-wall streaks in canonical wall-bounded turbulence-further supports the notion of this transition. Within the higher-shear patches, conditionally averaged statistics yield a log-variation in the local mean temperature profiles, in agreement with the log-law of the wall for mean temperature, and a $Ra^{0.37}$ effective power-law scaling of the local Nusselt number, consistent with the logarithmically corrected 1/2-power law scaling predicted for ultimate thermal convection for very large $Ra$. Collectively, the results from this study indicate that turbulent and laminar-like BL coexist in VC at moderate to high $Ra$ and this transition from the classical state to the ultimate state manifests as increasingly larger shear-dominated patches, consistent with the findings reported for Rayleigh-B\'enard convection and Taylor-Couette flows.