Temperature fluctuations near the ultimate-state transition in turbulent thermal convection

We report experimental results for temperature statistics in turbulent Rayleigh-Bénard convection (RBC) in a cylindrical sample of aspect ratio Γ ≡ D/L = 1.00 (D = 1.12m is the diameter and L = 1.12m the height). The measurements were conducted in the Rayleigh-number range 10^{13 <}_∼ Ra ^<_∼ 1.3 × 10¹⁴ and Pr ' 0.8. Using the elliptic approximation (EA), we derived an effective Reynolds number Re_eff from temperature space-time cross-correlation functions CT(z, τ). Results showed that Re_eff ∝ Ra^ζeff with ζ_eff = 0.44 ± 0.01 for Ra ^<_∼ 1.0 × 10¹⁴. At Ra ' 1.0 × 10¹⁴ we found a transition in the Ra dependence of Re_eff. At the largest Re measured, near Ra = 1.23 × 10¹⁴, we found ET(k) ∼ k^−1.67 for the temperature energy spectra, which agrees well with the Obukhov-Corrsin −5/3 scaling for passive scalars in a turbulent flow at sufficiently large Reynolds numbers.