Transition to ultimate Rayleigh-B\'{e}nard turbulence revealed through extended self similarity scaling analysis of the temperature structure functions

In turbulent Rayleigh-B\'{e}nard (RB) convection, a transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number $Ra \approx 10^{13}-10^{14}$ a transition in the scaling of the global Nusselt number $Nu$ (the dimensionless heat transfer) and the Reynolds number with $Ra$ has been observed in experiments and very recently in direct numerical simulations (DNS) of two-dimensional (2D) RB. In this paper we analyse the local scaling properties of the lateral temperature structure functions in the BLs of this simulation of 2D RB, employing extended self-similarity (ESS) (i.e., plotting the structure functions against each other, rather than only against the scale) in the spirit of the attached eddy hypothesis, as we have recently introduced for velocity structure functions in wall turbulence (Krug \textit{et al.}, \textit{J. Fluid Mech.}, vol. 830, 2017, pp. 797-819). 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, as expected for a scalar in a turbulent boundary layer. In striking correspondence to the $Nu$ scaling, the ESS scaling region is negligible at $Ra = 10^{11}$ and well developed at $Ra = 10^{14}$, thus providing strong evidence that the observed transition in the global Nusselt number at $Ra \approx 10^{13}$ indeed is the transition from a laminar type BL to a turbulent type BL. Our results further show that the relative slopes for scalar structure functions in the ESS scaling regime are the same as for their velocity counterparts extending their previously established universality. The findings are confirmed by comparing to scalar structure functions in 3D turbulent channel flow.