Transition to ultimate Rayleigh–Bénard turbulence revealed through extended self-similarity scaling analysis of the temperature structure functions

Abstract

In turbulent Rayleigh-Bénard (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 ≈ 10 13 − 10 14 a transition in the scaling of the global Nusselt number N u (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 et al., 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 N u 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 ≈ 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.

Zhu et al. Reply:

“…[2] has also been confirmed through an extended self-similarity (ESS) analysis of the temperature structure functions; see Ref. [5]. In that paper we find no ESS scaling before the transition.…”

Vertical Coherence of Turbulence in the Atmospheric Surface Layer: Connecting the Hypotheses of Townsend and Davenport

“…As a final remark, we point out that the above considerations made for the streamwise velocity component should also apply to the spanwise velocity field and temperature (Perry and Chong, 1982;Krug et al, 2018). Townsend's Davenport (1961a) presented a trend in the wall-normal coherence of streamwise velocity fluctuations u based on observations from typical tower micrometeorological data.…”

Reply to "Absence of Evidence for the Ultimate Regime in Two-Dimensional Rayleigh-Bénard Convection"

“…Finally, the transition to ultimate RB turbulence in the numerical data of [2] has also been confirmed through an extended self-similarity (ESS) analysis of the temperature structure functions, see ref. [5]: In that paper we find no ESS scaling before the transition. However, beyond the transition and for large enough wall distance y + > 100, we find clear ESS behaviour, as expected for a scalar in a turbulent boundary layer.…”