Local patches of turbulent boundary layer behaviour in classical-state vertical natural convection

We present evidence of local patches in vertical natural convection that are reminiscent of Prandtl--von K{\'a}rm{\'a}n turbulent boundary layers, for Rayleigh numbers $10^5$--$10^9$ and Prandtl number 0.709. These local patches exist in the classical state, where boundary layers exhibit a laminar-like Prandtl--Blasius--Polhausen scaling at the global level, and are distinguished by regions dominated by high shear and low buoyancy flux. Within these patches, the locally averaged mean temperature profiles appear to obey a log-law with the universal constants of Yaglom (1979). We find that the local Nusselt number versus Rayleigh number scaling relation agrees with the logarithmically corrected power-law scaling predicted in the ultimate state of thermal convection, with an exponent consistent with Rayleigh--B{\'e}nard convection and Taylor--Couette flows. The local patches grow in size with increasing Rayleigh number, suggesting that the transition from the classical state to the ultimate state is characterised by increasingly larger patches of the turbulent boundary layers.