Revealing ultimate Rayleigh-Bénard turbulence through scaling analysis of the temperature structure functions

The transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, in turbulent Rayleigh-Bénard convection (RBC) has been observed in some experiments and very recently in direct numerical simulations (DNS) of 2D RBC at Ra = 10¹³. Here, we make use of the same DNS to analyse the scaling properties of temperature structure functions S_p ≡ ⟨Δθ(r)^2p⟩^1/p in the BLs, where 2p is the (even) order and Δθ(r) the temperature difference across a lateral distance r. Based on e.g. the attached-eddy framework, it is possible to derive a scaling S_p(r) ∼ ln(r) for velocity structure functions in the logarithmic region. By employing extended self-similarity (ESS) ( i.e., plotting the structure functions against each other, rather than r), we have previously demonstrated that this scaling emerges even at low Reynolds numbers with universal relative slopes. In RBC, we find no ESS scaling below the transition and in the near wall region. However, beyond the transition and for large enough wall distance z⁺ > 100, we find clear ESS behaviour. Our analysis gives strong evidence that the observed transition in the global Nusselt number at Ra ≈ 10¹³ indeed is the transition from a laminar type BL to a turbulent type BL.